<Text-field style="Heading 1" layout="Heading 1">Initialization.</Text-field>

<Text-field style="Heading 1" layout="Heading 1">Review of Series and Power Series.</Text-field>

SERIES

Recall a power series in [powers of] x - a is an infinite series of the form

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

If a = 0, this is a power series in x

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

An important aspect of any series is whether or not it converges (IE the infinite sum exists). (one might say that the only good series is a converging series)

NOTE: A converging series can then be approximated by using its nth partial sum.

Definition 1. Given an infinite series 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 the sequence of partial sums of the series be 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. If this sequence is convergent IE if 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the series is called convergent and S is called its sum. If the limit 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does not exist, the series is called divergent.

Examples. 1. Geometric series 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where a and r are constants. This series converges to 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if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYoLUYjNiQtSSNtaUdGJDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6L0krbXNlbWFudGljc0dGJFEkYWJzRicvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR0ZBRjwtSSNtb0dGJDYtUSI8RidGOi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSi8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZZLUkjbW5HRiQ2JFEiMUYnRjpGOg== and diverges for 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 sum of the first n terms 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

2. p series 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where p is a constant, converges for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= and diverges for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJiZsZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIxRidGOUY5

<Text-field style="Heading 2" layout="Heading 2">Tests for convergence of a series</Text-field>

There are several tests for convergence of a series.

1. Comparison test for series of non-negative terms

Let 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and suppose LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSttdW5kZXJvdmVyR0YkNictSSNtb0dGJDYtUSYmU3VtO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGOy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y7LyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRjsvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xNjY2NjY3ZW1GJy1GIzYkRitGNkYrLyUnYWNjZW50R1EmZmFsc2VGJy8lLGFjY2VudHVuZGVyR0ZTRistRiM2JC1JJW1zdWJHRiQ2Ji1GLDYlUSJ2RicvJSdpdGFsaWNHRkAvRjdRJ2l0YWxpY0YnLUYjNiQtRiw2JVEibkYnRmhuRmpuRjYvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjZGK0Y2 converges. Then if 0\342\211\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 then 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 converges.

If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSttdW5kZXJvdmVyR0YkNictSSNtb0dGJDYtUSYmU3VtO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGOy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y7LyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRjsvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xNjY2NjY3ZW1GJy1GIzYkRitGNkYrLyUnYWNjZW50R1EmZmFsc2VGJy8lLGFjY2VudHVuZGVyR0ZTRistRiM2JC1JJW1zdWJHRiQ2Ji1GLDYlUSJ2RicvJSdpdGFsaWNHRkAvRjdRJ2l0YWxpY0YnLUYjNiQtRiw2JVEibkYnRmhuRmpuRjYvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjZGK0Y2 diverges and if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1GLzYjUSFGJy1GIzYkLUkjbW9HRiQ2LVEmJmdlcTtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZQLyUpc3RyZXRjaHlHRlAvJSpzeW1tZXRyaWNHRlAvJShsYXJnZW9wR0ZQLyUubW92YWJsZWxpbWl0c0dGUC8lJ2FjY2VudEdGUC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmluRj0tRiw2Ji1GLzYlUSJ2RidGMkY1RjhGP0ZCLUZLNi1RIixGJ0Y9Rk4vRlJGNEZTRlVGV0ZZRmVuL0ZoblEmMC4wZW1GJy9GW29RLDAuMzMzMzMzM2VtRictRks2LVEifkYnRj1GTkZRRlNGVUZXRllGZW5GZW8vRltvRmZvLUZLNi5RJGZvckYnLyUlYm9sZEdGUEY9Rk5GUUZTRlVGV0ZZRmVuRmVvL0Zbb1EsMC4yMjIyMjIyZW1GJ0Zpby1GLzYlUSRhbGxGJ0YyRjVGaW9GOi1GSzYtUSI+RidGPUZORlFGU0ZVRldGWUZlbkZnbkZqbi1GLzYlUSJORidGMkY1RmlvRj0= then 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 diverges.

2. Quotient test for series of non-negative terms.

If 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and 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and if 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either both converge or both diverge.

If A = 0 and if 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 then 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

If A = \342\210\236 and 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 then 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

3. Integral test. for series of non-negative terms.

If f(x) is positive, continuous and monotonic decreasing for x \342\211\245 N and is such that f(n) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y9, n = N, N+1, N+2, ...then 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 converges or diverges according as 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 converges or diverges.

4. Alternating Series test.

An alternating series converges if it satisfies the two conditions

a) 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 for n \342\211\245 1

b) 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or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Jy1JI21pR0YkNiVRIm5GJy8lJ2l0YWxpY0dGQi9GM1EnaXRhbGljRictRi82LVEtJnJpZ2h0YXJyb3c7RidGMkY1RjgvRjtGQkY8Rj4vRkFGN0ZDL0ZGUSwwLjI3Nzc3NzhlbUYnL0ZJRmVuLUYjNiQtRi82LVEoJiM4NzM0O0YnRjIvRjZRJmZhbHNlRicvRjlGXW8vRjtGXW8vRj1GXW8vRj9GXW8vRkFGXW8vRkRGXW9GRS9GSUZHRjItRk42I1EhRidGMi8lLGFjY2VudHVuZGVyR0Zdby1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUklbXN1YkdGJDYmLUZONiVRInVGJ0ZRRlMtRiM2JEZNRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjJGMi8lJW9wZW5HUSJ8Z3JGJy8lJmNsb3NlR0ZfcS1GLzYtUSI9RidGMkZcb0Zeb0Zfb0Zgb0Zhb0Zib0Zjb0ZaRmZuLUkjbW5HRiQ2JFEiMEYnRjItRi82LVEifkYnRjJGXG9GXm9GX29GYG9GYW9GYm9GY29GRUZkby1GLzYtUSIpRidGMi9GNkZCRl5vRlhGYG9GYW9GYm9GY28vRkZGSkZIRmlxRmlxRmlxRmlxRmlxRmlxRmlxRjI=

Definition: The series 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 is called absolutely convergent if 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. If 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 converges but 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 diverges, then 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 is called conditionally convergent.

NOTE: An absolutely convergent series is convergent.

5. Ratio test.

Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Jy1JI21pR0YkNiVRIm5GJy8lJ2l0YWxpY0dGQi9GM1EnaXRhbGljRictRi82LVEtJnJpZ2h0YXJyb3c7RidGMkY1RjgvRjtGQkY8Rj4vRkFGN0ZDL0ZGUSwwLjI3Nzc3NzhlbUYnL0ZJRmVuLUYjNiQtRi82LVEoJiM4NzM0O0YnRjIvRjZRJmZhbHNlRicvRjlGXW8vRjtGXW8vRj1GXW8vRj9GXW8vRkFGXW8vRkRGXW9GRS9GSUZHRjItRk42I1EhRidGMi8lLGFjY2VudHVuZGVyR0Zdby1JKG1mZW5jZWRHRiQ2KC1GIzYmRmVvLUYjNiQtSSZtZnJhY0dGJDYoLUklbXN1YkdGJDYmLUZONiVRInVGJ0ZRRlMtRiM2J0ZNLUYvNi1RIitGJ0YyRlxvRl5vRl9vRmBvRmFvRmJvRmNvL0ZGUSwwLjIyMjIyMjJlbUYnL0ZJRmBxLUYjNiQtSSNtbkdGJDYkUSIxRidGMkYyRmVvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUYjNiQtRmVwNiZGZ3AtRiM2JEZNRjJGaHFGW3JGMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaXIvJSliZXZlbGxlZEdGXW9GMkZlb0YyRjIvRlxyUSRhYnNGJy8lJW9wZW5HUSkmdmVyYmFyO0YnLyUmY2xvc2VHRmJzRl5zLUYvNi1RIj1GJ0YyRlxvRl5vRl9vRmBvRmFvRmJvRmNvRlpGZm4tRk42JVEiTEYnRlFGUy1GLzYtUSJ+RidGMkZcb0Zeb0Zfb0Zgb0Zhb0Zib0Zjb0ZFRmRvRjI= Then the series 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

a) converges (absolutely) if L < 1

b) diverges if L > 1

If L = 1 the test fails.

6. Raabe's test.

Let 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 Then the series 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

a) converges (absolutely) if L > 1

b) diverges or converges conditionally if L < 1

If L = 1 the test fails.

7. Gauss' test.

If 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where 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 for all n > N, then the series 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

a) converges absolutely if L > 1

b) diverges or is conditionally convergent if L \342\211\244 1

POWER SERIES

A power series 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 converges at the point x = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ if the infinite series (of real numbers) 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 converges.

In general a power series converges absolutely for 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and diverges for 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. R is called the radius of convergence.

Note: convergence at the endpoints requires separate analysis.

Example 1. Determine the convergence of 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

Applying the ratio test we have 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

Hence the series converges absolutely for 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diverges for 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

For 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 IE at 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we must test separately

At 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 our series is 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 the harmonic series known to diverge.

At x = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjdGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYkUSIyRidGMkYyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Y/LyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUkjbW9HRiQ2LVEifkYnRjIvJSZmZW5jZUdGRC8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZZRkVGMg==our series is 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 an alternating harmonic series that converges.

The given series converges for 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 and diverges everywhere else.

Definition A function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 is said to be analytic at a point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ if, in an open interval about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ this function can be represented as a power series that has a positive radius of convergence.

Taylor Series If f is analytic at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ , then the representation

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 holds in some open interval about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ and is called the Taylor series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= about 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 When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmVuRjpGPg== , it is also called the Maclaurin series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY5

<Text-field style="Heading 1" layout="Heading 1">Power Series Solutions to Linear Differential Equations.</Text-field>

We now consider a method for obtaining a power series solution to a linear differential equation with polynomial coefficients.

Given the differential equation

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

we begin by writing it in the standard form

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZcby1JI21pR0YkNiVRInlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSInRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnRjUtRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS0ZPRk0tRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZYLUYsNiVRInBGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGL0YyRjlGOUYrRjVGVC1GLDYlUSJxRidGL0YyRmduRistRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZmby1JI21uR0YkNiRRIjBGJ0Y5RlBGUEZQRlBGUEZQRlBGUEZQRlBGUEZQRlBGUEZQRlBGUEZQRlBGUEZQRlBGUEZQRlBGUEZQRlBGUEZQRlBGUEZQRlBGUC1GaG42JC1GIzYkLUZpbzYkUSIyRidGOUY5RjlGUEZQRlBGUEZQLUYsNiVRJndoZXJlRidGL0YyRlBGWkZnbkZiby1JJm1mcmFjR0YkNigtRiM2JS1JJW1zdWJHRiQ2Ji1GLDYlUSJhRidGL0YyLUYjNiQtRmlvNiRRIjFGJ0Y5RjkvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRmduRjktRiM2JS1GXHE2JkZecUZecEZmcUZpcUZnbkY5LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zlci8lKWJldmVsbGVkR0Y9LUYsNiNRIUYnRjk=and 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

Definition 1 A point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ is called an ordinary point of equation (1) if both p(x) and q(x) are analytic at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RIi5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVuRj4= If it is not an ordinary point, it is called a singular point of the equation.

Definition 2. A singular point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RIn5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVuRj4= of (2) is said to be a regular singular point if both 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 and 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 are analytic at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RIi5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVuRj4= Otherwise LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ is called an irregular singular point.

<Text-field style="Heading 2" layout="Heading 2">Solutions about an ordinary point x = <Equation executable="false" style="Heading 2" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEieEYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNictSSNtbkdGJDYlUSIwRidGMi9GOVEnbm9ybWFsRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjJGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRidGQ0ZGRjJGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEieEYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNictSSNtbkdGJDYlUSIwRidGMi9GOVEnbm9ybWFsRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjJGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRidGQ0ZGRjJGQQ==</Equation></Text-field>

We assume that a power series solution of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JK211bmRlcm92ZXJHRiQ2Jy1GNjYtUSYmU3VtO0YnRjkvRjxRJnVuc2V0RicvRj9GVi9GQUYxL0ZDRlYvRkVGMS9GR0YxL0ZJRlYvRktRJjAuMGVtRicvRk5RLDAuMTY2NjY2N2VtRictRiM2Ji1GLDYlUSJuRidGL0YyLUY2Ni1GOEY5RlVGVy9GQUZWRlkvRkVGVi9GR0ZWRmZuRkpGTS1JI21uR0YkNiRRIjBGJ0Y5RjktRiM2Ji1GLDYjUSFGJy1GIzYkLUY2Ni1RKCYjODczNDtGJ0Y5RjtGPkZARkJGREZGRkhGZ24vRk5GaG5GOUZbcEY5LyUnYWNjZW50R0Y9LyUsYWNjZW50dW5kZXJHRj0tSSVtc3ViR0YkNiYtRiw2JVEiYUYnRi9GMi1GIzYkRl1vRjkvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEieEYnRi9GMi1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GZXItRmlwNiZGXnItRiM2JEZlb0Y5RmBxRmNxRjlGOUZecS8lMXN1cGVyc2NyaXB0c2hpZnRHRmJxLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGZ25GY3BGOQ==exists and our task is to determine the coefficients This task is accomplished by substituting this series into the differential equation, combining the result into a single series by collecting the result in powers of x and then in order for this series to be identically zero, we must have that all of its coefficients must be equal to zero.

We make use of Maples commands , series,series ,coeff, coeff ,convert,convert collect, collect , etc to carry out the required work.

<Text-field style="Heading 3" layout="Heading 3">Example 1 Solve the equation 2y'' +xy' + y = 0 about the ordinary point x = 0.</Text-field>

Step 1. Define the deq

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

LywoKiYiIiMiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYoNiQtSSJ5RzYiNiNJInhHRi9GMUYxRiZGJiomRjFGJkYrRiZGJkYtRiYiIiE=

print();

LywoKiYiIiMiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYoNiQtSSJ5RzYiNiNJInhHRi9GMUYxRiZGJiomRjFGJkYrRiZGJkYtRiYiIiE=

Step 2. Define the series solution (begin by setting the order to be used for the series)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmT3JkZXJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOQ==

IiM1

print();

IiM1

KzlJInhHNiItSSJ5R0YkNiMiIiFGKC0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJkYnIiIiLCQqJiNGMCIiI0YwLS0tSSNAQEdGLDYkRitGNEYvRidGMEYwRjQsJComI0YwIiInRjAtLS1GODYkRisiIiRGL0YnRjBGMEZCLCQqJiNGMCIjQ0YwLS0tRjg2JEYrIiIlRi9GJ0YwRjBGSywkKiYjRjAiJD8iRjAtLS1GODYkRisiIiZGL0YnRjBGMEZULCQqJiNGMCIkPyhGMC0tLUY4NiRGK0Y9Ri9GJ0YwRjBGPSwkKiYjRjAiJVNdRjAtLS1GODYkRisiIihGL0YnRjBGMEZfbywkKiYjRjAiJj8uJUYwLS0tRjg2JEYrIiIpRi9GJ0YwRjBGaG8sJComI0YwIichKUdPRjAtLS1GODYkRisiIipGL0YnRjBGMEZhcC1JIk9HRi02I0YwIiM1

print();

Step 3. Substitute the series into the deq.

JSFH

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

print();

LywoKiYiIiMiIiIrNUkieEc2Ii0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRik2JEkiREdGLkYlNiNJInlHRik2IyIiIUY2LS0tRi02JEYyIiIkRjNGNUYmLCQqJiNGJkYlRiYtLS1GLTYkRjIiIiVGM0Y1RiZGJkYlLCQqJiNGJiIiJ0YmLS0tRi02JEYyIiImRjNGNUYmRiZGOywkKiYjRiYiI0NGJi0tLUYtNiRGMkZHRjNGNUYmRiZGQywkKiYjRiYiJD8iRiYtLS1GLTYkRjIiIihGM0Y1RiZGJkZMLCQqJiNGJiIkPyhGJi0tLUYtNiRGMiIiKUYzRjVGJkYmRkcsJComI0YmIiVTXUYmLS0tRi02JEYyIiIqRjNGNUYmRiZGZ24tSSJPR0YvNiNGJkZgb0YmRiYqJkYoRiYrN0YoLS1GMkYzRjVGNkYqRiYsJComRj5GJkY3RiZGJkYlLCQqJkZGRiZGP0YmRiZGOywkKiZGT0YmRkhGJkYmRkMsJComRldGJkZRRiZGJkZMLCQqJkZqbkYmRllGJkYmRkcsJComRmNvRiZGXG9GJkYmRmduLCQqJiNGJiImPy4lRiZGZW9GJkYmRmBvRmpvRmlvRiZGJis5RigtRjRGNUY2Rl9wRiYsJComRj5GJkYqRiZGJkYlLCQqJkZGRiZGN0YmRiZGOywkKiZGT0YmRj9GJkYmRkMsJComRldGJkZIRiZGJkZMLCQqJkZqbkYmRlFGJkYmRkcsJComRmNvRiZGWUYmRiZGZ24sJComRl9xRiZGXG9GJkYmRmBvLCQqJiNGJiInISlHT0YmRmVvRiZGJkZpb0ZqbyIjNUYmRjY=

Step 4 convert the series into a polynomial and collect in terms of powers of x

JSFH

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

print();

JSFH

Step 5. Use the Maple command coeff to select the coefficients of the powers of x. An example of this command is

JSFH

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

LCYqJiIiIyIiIi0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkiREdGKkYkNiNJInlHRi02IyIiIUYlRiUtRjFGMkYl

print();

LCYqJiIiIyIiIi0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkiREdGKkYkNiNJInlHRi02IyIiIUYlRiUtRjFGMkYl

We then set the coefficients equal to zero and solve for the unknown derivatives i terms of the initial conditions that are required in order to solve a second order DEQ IE y(0) = a and y'(0) = b

JSFH

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PCovLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiIqNiNJInlHRis2IyIiISwkKiYiI0MiIiJJImJHRitGNkY2Ly0tLUYnNiRGLSIiKUYvRjEsJComIyIkMCIiIztGNkkiYUdGK0Y2RjYvLS0tRic2JEYtIiIoRi9GMSwkKiYiIidGNkY3RjYhIiIvLS0tRic2JEYtRkxGL0YxLCQqJiMiIzpGPUY2RkNGNkZNLy0tLUYnNiRGLSIiJkYvRjEsJComIiIjRjZGN0Y2RjYvLS0tRic2JEYtIiIlRi9GMSwkKiYjIiIkRl9vRjZGQ0Y2RjYvLS0tRic2JEYtRmNvRi9GMSwkRjdGTS8tLS1GJzYkRi1GaW5GL0YxLCQqJiNGNkZpbkY2RkNGNkZN

print();

Step 6. Finally we substitute these values into our solution and collect the terms in terms of the parameters a and b.

JSFH

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LCYqJiwsIiIiRiUqJiNGJSIiJUYlKUkieEc2IiIiI0YlISIiKiYjRiUiI0tGJSlGKkYoRiVGJSomI0YlIiQlUUYlKUYqIiInRiVGLSomI0YlIiVXaEYlKUYqIiIpRiVGJUYlSSJhR0YrRiVGJSomLCxGKkYlKiYjRiVGNkYlKUYqIiIkRiVGLSomI0YlIiNnRiUpRioiIiZGJUYlKiYjRiUiJFMpRiUpRioiIihGJUYtKiYjRiUiJj9eIkYlKUYqIiIqRiVGJUYlSSJiR0YrRiVGJQ==

print();

We can check our result using Maple's dsolve command

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

Ly1JInlHNiI2I0kieEdGJSwmKiYsLCIiIkYrKiYjRisiIiVGKylGJyIiI0YrISIiKiYjRisiI0tGKylGJ0YuRitGKyomI0YrIiQlUUYrKUYnIiInRitGMSomI0YrIiVXaEYrKUYnIiIpRitGK0YrSSJhR0YlRitGKyomLCxGJ0YrKiYjRitGOkYrKUYnIiIkRitGMSomI0YrIiNnRispRiciIiZGK0YrKiYjRisiJFMpRispRiciIihGK0YxKiYjRisiJj9eIkYrKUYnIiIqRitGK0YrSSJiR0YlRitGKw==

print();

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<Text-field style="Heading 3" layout="Heading 3">Example 2. Solve the differential equation <Equation executable="false" style="Heading 3" input-equation="" display="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">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</Equation>about the ordinary point x = 0.</Text-field>

Step 1. Define the deq

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

LywoKiYsJiokKUkieEc2IiIiIyIiIkYrRitGK0YrLUklZGlmZkclKnByb3RlY3RlZEc2JC1GLTYkLUkieUdGKTYjRihGKEYoRitGK0YwISIiRjJGKyIiIQ==

print();

LywoKiYsJiokKUkieEc2IiIiIyIiIkYrRitGK0YrLUklZGlmZkclKnByb3RlY3RlZEc2JC1GLTYkLUkieUdGKTYjRihGKEYoRitGK0YwISIiRjJGKyIiIQ==

Step 2. Define the series solution (begin by setting the order to be used for the series)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmT3JkZXJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTQ==

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

print();

Step 3. Substitute the series into the equation

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

print();

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

Step 4 convert the series into a polynomial and collect in terms of powers of x

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSVlcW4xRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoY29sbGVjdEYnRjRGNy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRKXNpbXBsaWZ5RidGNEY3LUZYNiQtRiM2JC1GLDYlUShjb252ZXJ0RidGNEY3LUZYNiQtRiM2JUYxLUY7Ni1RIixGJ0Y+RkAvRkRGNkZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNRLDAuMzMzMzMzM2VtRictRiw2JVEocG9seW5vbUYnRjRGN0Y+Rj5GZG8tRiw2JVEieEYnRjRGN0Y+LUY7Ni1RIjtGJ0Y+RkBGZ29GRUZHRklGS0ZNRmhvRlJGKw==

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

print();

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Step 5. Use the Maple command coeff to select the coefficients of the powers of x. Then set the coefficients equal to zero and solve for the unknown derivatives i terms of the initial conditions that are required in order to solve a second order DEQ ie y(0) = a and y'(0) = b

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

PCovLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiInNiNJInlHRis2IyIiISwmKiYiI08iIiJJImJHRitGNkY2KiYiIzxGNkkiYUdGK0Y2ISIiLy0tLUYnNiRGLSIiJEYvRjEsJEY6RjsvLS0tRic2JEYtIiImRi9GMSwmKiZGQUY2RjdGNkY7KiYiIipGNkY6RjZGNi8tLS1GJzYkRi1GTEYvRjEsJiomIiV1X0Y2RjdGNkY7KiYiJXoiKkY2RjpGNkY2Ly0tLUYnNiRGLSIiKEYvRjEsJiomIiMqKkY2RjdGNkY2KiYiJDEjRjZGOkY2RjsvLS0tRic2JEYtIiIlRi9GMSwmRkpGOyomIiIjRjZGOkY2RjYvLS0tRic2JEYtIiIpRi9GMSwmKiYiJTw1RjZGN0Y2RjsqJiIkQCRGNkY6RjZGNi8tLS1GJzYkRi1GZG9GL0YxLCZGN0Y2RjpGOw==

print();

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Step 6. Finally we substitute these values into our solution and collect the terms in terms of the parameters a and b.

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

LCYqJiw0IiIiRiUqJiNGJSIiI0YlKUkieEc2IkYoRiUhIiIqJiNGJSIiJ0YlKUYqIiIkRiVGLComI0YlIiM3RiUpRioiIiVGJUYlKiYjRjEiI1NGJSlGKiIiJkYlRiUqJiMiIzwiJD8oRiUpRipGL0YlRiwqJiMiJC4iIiU/REYlKUYqIiIoRiVGLComIyIkMiIiJlNNIkYlKUYqIiIpRiVGJSomIyIleiIqIichKUdPRiUpRioiIipGJUYlRiVJImFHRitGJUYlKiYsMkYqRiVGJkYlKiYjRiVGTEYlRjVGJUYsKiYjRiVGOUYlRjpGJUYsKiYjRiUiIz9GJUZARiVGJSomIyIjNiIkZyZGJUZFRiVGJSomIyIkOCIiJSFbJUYlRktGJUYsKiYjIiQkSCImZywjRiVGUUYlRixGJUkiYkdGK0YlRiU=

print();

JSFH

We can check our result using Maple's dsolve command

Ly1JInlHNiI2I0kieEdGJSwmKiYsNCIiIkYrKiYjRisiIiNGKylGJ0YuRishIiIqJiNGKyIiJ0YrKUYnIiIkRitGMComI0YrIiM3RispRiciIiVGK0YrKiYjRjUiI1NGKylGJyIiJkYrRisqJiMiIzwiJD8oRispRidGM0YrRjAqJiMiJC4iIiU/REYrKUYnIiIoRitGMComIyIkMiIiJlNNIkYrKUYnIiIpRitGKyomIyIleiIqIichKUdPRispRiciIipGK0YrRitJImFHRiVGK0YrKiYsMkYnRitGLEYrKiYjRitGUEYrRjlGK0YwKiYjRitGPUYrRj5GK0YwKiYjRisiIz9GK0ZERitGKyomIyIjNiIkZyZGK0ZJRitGKyomIyIkOCIiJSFbJUYrRk9GK0YwKiYjIiQkSCImZywjRitGVUYrRjBGK0kiYkdGJUYrRis=

print();

JSFH

NOTE: One of the difficulties is determining the radius of convergence when we are not able to find a general form for the coefficients and use the ratio test.

EXISTENCE OF ANALYTIC SOLUTIONS.

There is a theorem that states that if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RIn5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVuRj4=is an ordinary point then equation has two linearly independent analytic solutions of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkrbXVuZGVyb3ZlckdGJDYnLUZANi1RJiZTdW07RidGPS9GRFEmdW5zZXRGJy9GR0Zobi9GSUYxL0ZLRmhuL0ZNRjEvRk9GMS9GUUZobi9GU1EmMC4wZW1GJy9GVlEsMC4xNjY2NjY3ZW1GJy1GIzYmLUYsNiVRIm5GJ0YvRjItRkA2LUZCRj1GZ25GaW4vRklGaG5GW28vRk1GaG4vRk9GaG5GXm9GUkZVLUkjbW5HRiQ2JFEiMEYnRj1GPS1GIzYmLUYsNiNRIUYnLUYjNiQtRkA2LVEoJiM4NzM0O0YnRj1GQ0ZGRkhGSkZMRk5GUEZfby9GVkZgb0Y9RmNwRj0vJSdhY2NlbnRHRkUvJSxhY2NlbnR1bmRlckdGRS1JJW1zdWJHRiQ2Ji1GLDYlUSJhRidGL0YyLUYjNiRGZW9GPS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictSSVtc3VwR0YkNiUtRjY2JC1GIzYmRjotRkA2LVEqJnVtaW51czA7RidGPUZDRkZGSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRmlyLUZhcTYmRjotRiM2JEZdcEY9RmhxRltyRj1GPUZmcS8lMXN1cGVyc2NyaXB0c2hpZnRHRmpxLUZANi1RIn5GJ0Y9RkNGRkZIRkpGTEZORlBGX29GW3FGPQ== and that the radius of convergence is as least as large as the distance from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ to the nearest singular point (real or complex-valued) of the deq.

In example 1 above since there are no singular points the radius of convergence is \342\210\236.

In example 2 there are two singular points \302\261 i and hence the radius of convergence of our solutions about the point x = 0 is at least 1.

Maple also has a power series package PowerSeries that allows you to work directly with power series.

We will use this package in the following examples.

<Text-field style="Heading 3" layout="Heading 3">Example 1. Solve the IVP y'' - (x-2)y' + 2y = 0; y(0) = 1, y'(0) = -1</Text-field>

We will use power series package in Maple to find the solution

First to create the series solution Ys(x) = 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

The command tpsform converts the Powseries created above into a power series form of the variable stated in the command.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEodHBzZm9ybUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRI1lzRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJ4RidGL0YyRj0tSSNtbkdGJDYkUSMxMEYnRkFGQQ==

KzlJInhHNiImSSJhR0YkNiMiIiFGKCZGJjYjIiIiRismRiY2IyIiI0YuJkYmNiMiIiRGMSZGJjYjIiIlRjQmRiY2IyIiJkY3JkYmNiMiIidGOiZGJjYjIiIoRj0mRiY2IyIiKUZAJkYmNiMiIipGQy1JIk9HJSpwcm90ZWN0ZWRHRioiIzU=

print();

KzlJInhHNiImSSJhR0YkNiMiIiFGKCZGJjYjIiIiRismRiY2IyIiI0YuJkYmNiMiIiRGMSZGJjYjIiIlRjQmRiY2IyIiJkY3JkYmNiMiIidGOiZGJjYjIiIoRj0mRiY2IyIiKUZAJkYmNiMiIipGQy1JIk9HJSpwcm90ZWN0ZWRHRioiIzU=

Next we determine the necessary derivatives found in the given deq.

JSFH

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

print();

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

print();

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

Note the answer is in the form of a procedure. You can see the power series by using the tpsform command.

We also need to make the coeff of our eqn into a power series in order to use the powseries commands.

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

You can see the power series by using the tpsform command. For example

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEodHBzZm9ybUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRI2MxRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJ4RidGL0YyRkE=

KydJInhHNiIiIiMiIiEhIiIiIiI=

print();

KydJInhHNiIiIiMiIiEhIiIiIiI=

Next we combine these series to form the lhs of the equation given using the commands for multiplying and adding power series.

JSFH

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

print();

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

KzlJInhHNiIsKComIiIjIiIiJkkiYUdGJDYjRidGKEYoKiZGJ0YoJkYqNiNGKEYoRigqJkYnRigmRio2IyIiIUYoRihGMiwoKiYiIidGKCZGKjYjIiIkRihGKEYtRigqJiIiJUYoRilGKEYoRigsJiomIiM3RigmRio2I0Y6RihGKEY0RihGJywoKiYiIz9GKCZGKjYjIiImRihGKEY2ISIiKiYiIilGKEY+RihGKEY4LCgqJiIjSUYoJkYqNiNGNUYoRigqJkYnRihGPkYoRkYqJiIjNUYoRkNGKEYoRjosKComIiNVRigmRio2IyIiKEYoRigqJkY4RihGQ0YoRkYqJkY9RihGTEYoRihGRSwoKiYiI2NGKCZGKjYjRkhGKEYoKiZGOkYoRkxGKEZGKiYiIzlGKEZURihGKEY1LCgqJiIjc0YoJkYqNiMiIipGKEYoKiZGRUYoRlRGKEZGKiYiIztGKEZmbkYoRihGViwoKiYiIyEqRigmRio2I0ZQRihGKComRjVGKEZmbkYoRkYqJiIjPUYoRl5vRihGKEZILCgqJiIkNSJGKCZGKjYjIiM2RihGKComRlZGKEZeb0YoRkYqJkZCRihGZ29GKEYoRmBvLUkiT0clKnByb3RlY3RlZEdGLkZQ

print();

Using the op command to identify the coefficients , setting them equal to zero, and solving for them in terms of a[0] and a[1]

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

print();

Substituting the coeff into Y(x)

JSFH

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

KzlJInhHNiImSSJhR0YkNiMiIiFGKCZGJjYjIiIiRissJkYpISIiRiVGLSIiIywmKiYjRitGLkYrRilGK0YrKiYjRi4iIiRGK0YlRitGK0Y0LCYqJiNGKyIiJUYrRilGK0YtKiYjRitGNEYrRiVGK0YtRjgsJiomI0YrIiIpRitGKUYrRisqJiNGKyIiJ0YrRiVGK0YrIiImLCYqJiMiIigiJD8iRitGKUYrRi0qJiNGRiIjISpGK0YlRitGLUZBLCYqJiMiI1YiJSFvIkYrRilGK0YrKiYjRk4iJWc3RitGJUYrRitGRiwmKiYjIiNyIiU/bkYrRilGK0YtKiYjRlYiJVNdRitGJUYrRi1GPiwmKiYjIiQqXCInZzQ3RitGKUYrRisqJiNGaG4iJj8yKkYrRiVGK0YrIiIqLUkiT0clKnByb3RlY3RlZEdGKiIjNQ==

print();

JSFH

Converting our solution into a polynomial and collecting in terms of

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

M0ljbyNtc3ViKG1pKCJhIixmb250c3R5bGV+PX4iaXRhbGljIiksbXJvdyhtbigiMCIpLG1vKCImSW52aXNpYmxlVGltZXM7IikpKUc2IklMI21zdWIobWkoImEiLGZvbnRzdHlsZX49fiJpdGFsaWMiKSxtbigiMSIpKUdGJA==

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

print();

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

LCYqJiw0IiIiRiUqJClJInhHNiIiIiNGJSEiIiomI0YqIiIkRiUpRihGLkYlRiUqJiNGJUYuRiUpRigiIiVGJUYrKiYjRiUiIidGJSlGKCIiJkYlRiUqJiMiIigiIyEqRiUpRihGNkYlRisqJiMiI1YiJWc3RiUpRihGO0YlRiUqJiMiI3IiJVNdRiUpRigiIilGJUYrKiYjIiQqXCImPzIqRiUpRigiIipGJUYlRiUmSSJhR0YpNiMiIiFGJUYlKiYsNEYoRiVGJkYrKiYjRiVGKkYlRi9GJUYlKiYjRiVGM0YlRjJGJUYrKiYjRiVGSEYlRjdGJUYlKiYjRjsiJD8iRiVGPUYlRisqJiNGQCIlIW8iRiVGQkYlRiUqJiNGRSIlP25GJUZHRiVGKyomI0ZLIidnNDdGJUZNRiVGJUYlJkZQNiNGJUYlRiU=

print();

Finally, substituting the initial conditions and solving for a[0] and a[1]

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

PCQvJkkiYUc2IjYjIiIhIiIiLyZGJTYjRikhIiI=

print();

PCQvJkkiYUc2IjYjIiIhIiIiLyZGJTYjRikhIiI=

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

LDQiIiJGI0kieEc2IiEiIiomI0YjIiInRiMpRiQiIiRGI0YjKiYjRiMiIzdGIylGJCIiJUYjRiYqJiNGIyIjQ0YjKUYkIiImRiNGIyomIyIiKCIkZyRGIylGJEYpRiNGJiomIyIjViIlU11GIylGJEY4RiNGIyomIyIjciImZywjRiMpRiQiIilGI0YmKiYjIiQqXCInISlHT0YjKUYkIiIqRiNGIw==

print();

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

Using Maples dsolve command to solve the given deq.

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

Ly1JInlHNiI2I0kieEdGJSs3RiciIiIiIiEhIiJGKSNGKSIiJyIiJCNGKyIjNyIiJSNGKSIjQyIiJiMhIigiJGckRi0jIiNWIiVTXSIiKCMhI3IiJmcsIyIiKSMiJCpcIichKUdPIiIqLUkiT0clKnByb3RlY3RlZEc2I0YpIiM1

print();

Ly1JInlHNiI2I0kieEdGJSs3RiciIiIiIiEhIiJGKSNGKSIiJyIiJCNGKyIjNyIiJSNGKSIjQyIiJiMhIigiJGckRi0jIiNWIiVTXSIiKCMhI3IiJmcsIyIiKSMiJCpcIichKUdPIiIqLUkiT0clKnByb3RlY3RlZEc2I0YpIiM1

NOTE: Since this deq has NO singular points the radius of convergence for this solution is \342\210\236. ie it is valid for all values of x.

<Text-field style="Heading 3" layout="Heading 3">Example 2. Solve the DEQ y'' + xy' +(2x-3)y = 0 near x = -1.</Text-field>

Note x = -1 is an ordinary point for this deq.

We first make the substitution t = x - (-1) = x + 1. The resulting deq can then be solved near t = 0 using the above techniques.

ie Since y(x) = y(t-1) and 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and 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the eqn becomes 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

First create the power series solution

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

To calculate the derivative of Y(x)

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

print();

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

print();

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

converting the coefficients into power series

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

Combining these series as defined by the equation

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlElc2xuMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGMi9GNlEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUYsNiZRJ3Bvd2FkZEYnRi9GMkY1LUkobWZlbmNlZEdGJDYkLUYjNiktRiw2JVEkZGR5RidGL0Y1LUY5Ni1RIixGJ0Y8Rj4vRkFGMUZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBRLDAuMzMzMzMzM2VtRictRiw2JVEpbXVsdGlwbHlGJ0YvRjUtRlU2JC1GIzYlLUYsNiVRI2MxRidGL0Y1RmZuLUYsNiVRI2R5RidGL0Y1RjxGZm5GXm8tRlU2JC1GIzYlLUYsNiVRI2MyRidGL0Y1RmZuLUYsNiVRI1lzRidGL0Y1RjxGPC1GOTYuUSI6RidGMkY8Rj5GQEZCRkRGRkZIRkpGTEZP

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

print();

Using the op command to identify the coefficients , setting them equal to zero, and solving for them in terms of a[0] and a[1]

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

print();

Substituting the coeff into Y(x) and converting the solution into a polynomial

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlElc29sbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGMi9GNlEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUYsNiZRJXNvcnRGJ0YvRjJGNS1JKG1mZW5jZWRHRiQ2JS1GIzYoLUYsNiZRKGNvbGxlY3RGJ0YvRjJGNS1GVTYlLUYjNiYtRiw2JlEoY29udmVydEYnRi9GMkY1LUZVNiUtRiM2Ji1GLDYmUSVzdWJzRidGL0YyRjUtRlU2JC1GIzYmLUYsNiVRJWNvZWZGJ0YvRjUtRjk2LVEiLEYnRjxGPi9GQUYxRkJGREZGRkhGSi9GTVEmMC4wZW1GJy9GUFEsMC4zMzMzMzMzZW1GJy1GLDYlUSh0cHNmb3JtRidGL0Y1LUZVNiQtRiM2Jy1GLDYlUSNZc0YnRi9GNUZbcC1GLDYlUSJ0RidGL0Y1RltwLUkjbW5HRiQ2JFEjMTBGJ0Y8RjxGPEZbcC1GLDYlUShwb2x5bm9tRidGL0Y1RjJGPEZbcC1GVTYnLUYjNictRiw2JlEiYUYnRi9GMkY1LUZVNictRiM2Iy1GYXE2JVEiMEYnRjJGPEYyRjwvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjk2LkZdcEYyRjxGPkZecEZCRkRGRkZIRkpGX3BGYXBGW3ItRlU2Jy1GIzYjLUZhcTYlUSIxRidGMkY8RjJGPEZlckZockYyRjwvRmZyUSJ8ZnJGJy9GaXJRInxockYnRjJGPEZbcy1GLDYmRl9xRi9GMkY1RltzLUYsNiZRKmFzY2VuZGluZ0YnRi9GMkY1RjJGPA==

LCYqJiw0IiIiRiUqJiMiIiYiIiNGJSlJInRHNiJGKUYlRiUqJiNGJUYpRiUpRisiIiRGJUYlKiYjRjAiIiVGJSlGK0YzRiVGJSomI0YlIiM/RiUpRitGKEYlISIiKiYjRiUiI2dGJSlGKyIiJ0YlRjkqJiNGMyIkMCJGJSlGKyIiKEYlRjkqJiNGMCIlPzZGJSlGKyIiKUYlRjkqJiMiI1AiJlMtJEYlKUYrIiIqRiVGJUYlJkkiYUdGLDYjIiIhRiVGJSomLDRGK0YlKiZGLkYlRipGJUYlKiYjRihGPkYlRi9GJUYlKiYjRiVGPkYlRjRGJUYlKiYjRiUiIzpGJUY4RiVGJSomI0ZDIiQhPUYlRj1GJUY5KiYjIiM8IiVnN0YlRkJGJUY5KiYjRlxvIiVTXUYlRkdGJUY5KiYjRkMiJSFbJ0YlRk1GJUYlRiUmRlA2I0YlRiVGJQ==

print();

Substituting that t = x+1 we obtain the solution to the original problem

JSFH

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

LCYqJiw0IiIiRiUqJiMiIiYiIiNGJSksJkkieEc2IkYlRiVGJUYpRiVGJSomI0YlRilGJSlGKyIiJEYlRiUqJiNGMSIiJUYlKUYrRjRGJUYlKiYjRiUiIz9GJSlGK0YoRiUhIiIqJiNGJSIjZ0YlKUYrIiInRiVGOiomI0Y0IiQwIkYlKUYrIiIoRiVGOiomI0YxIiU/NkYlKUYrIiIpRiVGOiomIyIjUCImUy0kRiUpRisiIipGJUYlRiUmSSJhR0YtNiMiIiFGJUYlKiYsNkYsRiVGJUYlKiZGL0YlRipGJUYlKiYjRihGP0YlRjBGJUYlKiYjRiVGP0YlRjVGJUYlKiYjRiUiIzpGJUY5RiVGJSomI0ZEIiQhPUYlRj5GJUY6KiYjIiM8IiVnN0YlRkNGJUY6KiYjRl1vIiVTXUYlRkhGJUY6KiYjRkQiJSFbJ0YlRk5GJUYlRiUmRlE2I0YlRiVGJQ==

print();

Again since this eqn has NO singular points this solution is valid for all values of x.

Given initial conditions we could then proceed to determine the values for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RIn5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVuRj4= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RIn5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVuRkZGPg==

Suppose that y(-1) = 2 and y'(-1) = -2

JSFH

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

PCQvJkkiYUc2IjYjIiIhIiIjLyZGJTYjIiIiISIj

print();

PCQvJkkiYUc2IjYjIiIhIiIjLyZGJTYjIiIiISIj

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

LDQqJiIiJSIiIiksJkkieEc2IkYlRiVGJSIiI0YlRiUqJiNGKiIiJEYlKUYnRi1GJSEiIiomIyIiKCIiJ0YlKUYnRiRGJUYlKiYjRjIiI0lGJSlGJyIiJkYlRi8qJiNGKiIjWEYlKUYnRjNGJUYlKiYjIiNKIiRJJ0YlKUYnRjJGJUYvKiYjRiUiJD8oRiUpRiciIilGJUYlKiYjIiM4IiZnYCVGJSlGJyIiKkYlRiUqJkYqRiVGKEYlRi8=

print();

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

Using Maple's dsolve command.

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

Ly1JInlHNiI2I0kieEdGJSs9LCZGJyIiIkYqRioiIiMiIiEhIiNGKiIiJUYrI0YtIiIkRjAjIiIoIiInRi4jISIoIiNJIiImI0YrIiNYRjMjISNKIiRJJ0YyI0YqIiQ/KCIiKSMiIzgiJmdgJSIiKiMiI2giJituJiIjNSMiI0oiJ2cqKVwiIzYtSSJPRyUqcHJvdGVjdGVkRzYjRioiIzc=

print();

JSFH

<Text-field style="Heading 2" layout="Heading 2">Solutions about a regular singular point --- Method of Frobenius.</Text-field>

Definition 3. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ is a regular singular point of y'' + p(x) y' +q(x) y = 0, then the indicial equation for this point is

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

where 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

The roots of the indicial equation are called the exponents(indices) of the singularity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+

Frobenius method of solving ordinary differential equations near a regular singular point, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+, by positing a solution of the form

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

the values of r and the coefficients LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RIn5GJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZLLyUpc3RyZXRjaHlHRksvJSpzeW1tZXRyaWNHRksvJShsYXJnZW9wR0ZLLyUubW92YWJsZWxpbWl0c0dGSy8lJ2FjY2VudEdGSy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlpGPQ== are then found by iteration by substituting the potential solution into the equation.

NOTE The first step in this method is to find the roots LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ (Re 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) of the indicial equation. Then utilizing the larger root Frobenius's theorem assures us that our deq has a series solution of the form above and that this series converges for all x such that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjxGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYzNi1RKiZ1bWludXMwO0YnRi9GNkY5RjtGPUY/RkFGQy9GRlEsMC4yMjIyMjIyZW1GJy9GSUZXLUklbXN1YkdGJDYmRkotRiM2JEYrRi8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjItRks2JVEiUkYnRk5GUS1GMzYtUSIsRidGL0Y2L0Y6RlBGO0Y9Rj9GQUZDL0ZGUSYwLjBlbUYnL0ZJUSwwLjMzMzMzMzNlbUYnLUYzNi1RIn5GJ0YvRjZGOUY7Rj1GP0ZBRkNGZW8vRklGZm9GLw==where R is the distance from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y+ to the nearest other singular point (real or complex).

If 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 is not an integer, then there exist two linearly independent solutions of the form

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

If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmVuLUYsNiZGLi1GIzYkLUY7NiRRIjJGJ0Y+Rj5GQEZDLUZHNi1RIn5GJ0Y+RkpGTUZPRlFGU0ZVRlcvRlpRJjAuMGVtRicvRmduRmNvLUZHNi1RIixGJ0Y+RkovRk5GNEZPRlFGU0ZVRldGYm8vRmduUSwwLjMzMzMzMzNlbUYnRl9vRj4=then there exist two linearly independent solutions of the form

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

If 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 is a positive integer, then there exist two linearly independent solutions of the form

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where C is a constant that could be zero.

<Text-field style="Heading 3" layout="Heading 3">Example 1. Find a series solution for the differential equation <Equation executable="false" style="Heading 3" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZMLUkobWZlbmNlZEdGJDYmLUYjNiotSSNtaUdGJDYlUSJ4RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYvUSIrRidGNC8lJ2l0YWxpY0dGNi9GOFEnbm9ybWFsRicvJSZmZW5jZUdGNi8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0Y2LyUobGFyZ2VvcEdGNi8lLm1vdmFibGVsaW1pdHNHRjYvJSdhY2NlbnRHRjYvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZSLUkjbW5HRiQ2JlEiMkYnRjRGPkZALyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0Y0Rj5GQEY0Rj5GQC1GOzYvUScmc2RvdDtGJ0Y0Rj5GQEZCRkRGRkZIRkpGTEZOL0ZRUSYwLjBlbUYnL0ZURl1vLUklbXN1cEdGJDYlRjAtRiM2KEZVRllGZm5GNEY+RkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYxNiVRInlGJ0Y0RjctRjs2L1EiJ0YnRjRGPkZARkJGREZGRkhGSkZMRk4vRlFRLDAuMTExMTExMWVtRidGXm9Gam8tRjs2L1EqJnVtaW51czA7RidGNEY+RkBGQkZERkZGSEZKRkxGTkZQRlNGMEZpbkZnb0Zqb0Y6LUYsNiYtRiM2Ki1GVjYmUSIxRidGNEY+RkBGOkYwRllGZm5GNEY+RkBGNEY+RkBGaW5GZ28tRjs2L1EiPUYnRjRGPkZARkJGREZGRkhGSkZMRk4vRlFRLDAuMjc3Nzc3OGVtRicvRlRGXXEtRlY2JlEiMEYnRjRGPkZALUY7Ni9RIixGJ0Y0Rj5GQEZCL0ZFUSV0cnVlRidGRkZIRkpGTEZORlxvL0ZUUSwwLjMzMzMzMzNlbUYnLUY7Ni9RIn5GJ0Y0Rj5GQEZCRkRGRkZIRkpGTEZORlxvRl5vRmlxRjAtRjs2L1EiPkYnRjRGPkZARkJGREZGRkhGSkZMRk5GXHFGXnFGX3FGaXFGaXFGaXEtRjE2JVEmYWJvdXRGJ0Y0RjdGaXEtRjE2JVEkdGhlRidGNEY3RmlxLUYxNiVRJnBvaW50RidGNEY3RmlxRjBGaXFGaXBGaXFGX3FGWUZmbkY0Rj5GQA==</Equation> </Text-field>

Now x = 0 and x = -2 are both singular points for this deq.

Also x = 0 is a regular singular point since 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 are analytic at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=

We are looking for a solution of the form 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

Step 1. Define our deq

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

LywoKigsJkkieEc2IiIiIiIiI0YoRigpRiZGKUYoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GLDYkLUkieUdGJzYjRiZGJkYmRihGKComRiZGKEYvRighIiIqJiwmRihGKEYmRihGKEYxRihGKCIiIQ==

print();

LywoKigsJkkieEc2IiIiIiIiI0YoRigpRiZGKUYoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GLDYkLUkieUdGJzYjRiZGJkYmRihGKComRiZGKEYvRighIiIqJiwmRihGKEYmRihGKEYxRihGKCIiIQ==

JSFH

Step 2. Define our series solution

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

print();

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

print();

Step 3. Substitute our series into the deq.

JSFH

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

print();

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

JSFH

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

print();

JSFH

Step 4. Determine the coefficient of the term LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JC1GLzYnUSJyRidGMkY1RjdGOi9GOFEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkI= ( it in effect is the indicial eqn needed to determine the values of r in Frobenius' method )

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnZmFjdG9yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmY29lZmZGJ0YvRjItRjY2JC1GIzYmLUYsNiVRJGxoc0YnRi9GMi1GNjYkLUYjNiMtRiw2JVElZXFuMUYnRi9GMi9GM1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0ZLLyUmZmVuY2VHRlMvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGUy8lKnN5bW1ldHJpY0dGUy8lKGxhcmdlb3BHRlMvJS5tb3ZhYmxlbGltaXRzR0ZTLyUnYWNjZW50R0ZTLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSVtc3VwR0YkNiUtRiw2JlEieEYnRi9GUUYyLUYjNiMtRiw2JlEickYnRi9GUUYyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZLRkstRk42LVEiO0YnRktGVEZWRlhGWkZmbkZobkZqbkZcby9GYG9RLDAuMjc3Nzc3OGVtRic=

KigtSSN5MUc2IjYjIiIhIiIiLCYqJiIiI0YoSSJyR0YlRihGKEYoISIiRigsJkYsRihGKEYtRig=

print();

KigtSSN5MUc2IjYjIiIhIiIiLCYqJiIiI0YoSSJyR0YlRihGKEYoISIiRigsJkYsRihGKEYtRig=

From the above we see that r = 1 or r = 1/2

Step 5. We use the larger value of r = 1 first. We now substitute that y1(0) = a , D(y1)(0) = b and r =1 into the coefficients of the lhs our eqn1

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

print();

JSFH

We now want to solve the above coefficients = 0 for the values of the derivatives in each of them

To do this we must drop the first and 2nd items. NOTE : the 2nd item a + 3b =0 says that b = -1/3 a

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

PCkvLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiImNiNJI3kxR0YrNiMiIiEsJComIyIjIioiI2IiIiJJImJHRitGOEY4Ly0tLUYnNiRGLSIiJEYvRjEsJComI0Y/Ri5GOEY5RjhGOC8tLS1GJzYkRi0iIiVGL0YxLCQqJiMiIzgiIzpGOEY5RjghIiIvLS0tRic2JEYtIiIjRi9GMSwkRkFGTi8tLS1GJzYkRi0iIihGL0YxLCQqJiMiJUokKiIkRClGOEY5RjhGOC8tLS1GJzYkRi0iIidGL0YxLCQqJiMiJDwjRjdGOEY5RjhGTi8tLS1GJzYkRi0iIilGL0YxLCQqJiMiJypHeCIiJXZZRjhGOUY4Rk4=

print();

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

PCkvLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiImNiNJI3kxR0YrNiMiIiEsJComIyIjIioiJGwiIiIiSSJhR0YrRjghIiIvLS0tRic2JEYtIiIkRi9GMSwkKiYjRjhGLkY4RjlGOEY6Ly0tLUYnNiRGLSIiJUYvRjEsJComIyIjOCIjWEY4RjlGOEY4Ly0tLUYnNiRGLSIiI0YvRjEsJEZCRjgvLS0tRic2JEYtIiIoRi9GMSwkKiYjIiVKJCoiJXZDRjhGOUY4RjovLS0tRic2JEYtIiInRi9GMSwkKiYjIiQ8I0Y3RjhGOUY4RjgvLS0tRic2JEYtIiIpRi9GMSwkKiYjIicqR3giIiZEUyJGOEY5RjhGOA==

print();

Step 6. Substitute these values into slna minus the last three terms since we have not calculated their correct coefficients

JSFH

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

KiYsNEkieEc2IiIiIiomI0YmIiIkRiYpRiQiIiNGJiEiIiomI0YmIiM1RiYpRiRGKUYmRiYqJiNGJiIjSUYmKUYkIiIlRiZGLComIyIjOCIlITMiRiYpRiQiIiZGJkYmKiYjIiMiKiImKyk+RiYpRiQiIidGJkYsKiYjIiQ8IyInKyk9IkYmKUYkIiIoRiZGJiomIyIlTDgiKCs/eSJGJilGJCIiKUYmRiwqJiMiJkZgIyIpK1N5ISlGJilGJCIiKkYmRiZGJkkiYUdGJUYm

print();

JSFH

It may be possible to obtain the second independent series solution by repeating the above using the other value of r ( This will work in this example since the two r values do not differ by an integer).

JSFH

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

print();

JSFH

We now want to solve the above coefficients = 0 for the values of the derivatives in each of them

To do this we must drop the first and 2nd items. NOTE : 2nd item says b=-3/4a

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

PCkvLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiIlNiNJI3kxR0YrNiMiIiEsJComIyIkWiMiJD8kIiIiSSJiR0YrRjghIiIvLS0tRic2JEYtIiImRi9GMSwkKiYjIiZcbCIiJj86IkY4RjlGOEY4Ly0tLUYnNiRGLSIiI0YvRjEsJComIyIiKCIjN0Y4RjlGOEY6Ly0tLUYnNiRGLSIiJEYvRjEsJComIyIkTCIiJFMjRjhGOUY4RjgvLS0tRic2JEYtIiInRi9GMSwkKiYjIihaWHEiIichKW9dRjhGOUY4RjovLS0tRic2JEYtIiIpRi9GMSwkKiYjIitwOGF5NyIpKy9iU0Y4RjlGOEY6Ly0tLUYnNiRGLUZPRi9GMSwkKiYjIihKW1UnIidTZW5GOEY5RjhGOA==

print();

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmY2ZmMmFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSomY29sb25lcTtGJ0YvRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi1GLDYlUSVzdWJzRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiwtRjY2LVEifkYnL0YzUSdub3JtYWxGJ0Y5RjxGPkZARkJGREZGL0ZJUSYwLjBlbUYnL0ZMRmVuLUYsNiVRImJGJ0YvRjItRjY2LVEiPUYnRlhGOUY8Rj5GQEZCRkRGRkZIRktGVS1GNjYtUSomdW1pbnVzMDtGJ0ZYRjlGPEY+RkBGQkZERkYvRklRLDAuMjIyMjIyMmVtRicvRkxGYW8tSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiM0YnRlgtRiM2Iy1GZ282JFEiNEYnRlgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmRwLyUpYmV2ZWxsZWRHRjstRjY2LVEnJnNkb3Q7RidGWEY5RjxGPkZARkJGREZGRlpGZm4tRiw2JVEiYUYnRi9GMi1GNjYtUSIsRidGWEY5L0Y9RjFGPkZARkJGREZGRlovRkxRLDAuMzMzMzMzM2VtRictRiw2JVElY2ZmMkYnRi9GMkZY

PCkvLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiIlNiNJI3kxR0YrNiMiIiEsJComIyIkVCgiJSFHIiIiIkkiYUdGK0Y4RjgvLS0tRic2JEYtIiIpRi9GMSwkKiYjIitwOGF5NyIpK3MxYUY4RjlGOEY4Ly0tLUYnNiRGLSIiKEYvRjEsJComIyIoSltVJyInPzYhKkY4RjlGOCEiIi8tLS1GJzYkRi0iIiRGL0YxLCQqJiMiJEwiIiQ/JEY4RjlGOEZQLy0tLUYnNiRGLSIiJ0YvRjEsJComIyIoWlhxIiInU2VuRjhGOUY4RjgvLS0tRic2JEYtIiImRi9GMSwkKiYjIiZcbCIiJmdgIkY4RjlGOEZQLy0tLUYnNiRGLSIiI0YvRjEsJComI0ZKIiM7RjhGOUY4Rjg=

print();

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Finally we substitute these values into slna minus the last three terms since we have not calculated their correct coefficients

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

KiYsNCokKUkieEc2IiMiIiIiIiNGKUYpKiYjIiIkIiIlRikpRiYjRi1GKkYpISIiKiYjIiIoIiNLRikpRiYjIiImRipGKUYpKiYjIiRMIiIlPz5GKSlGJiNGNEYqRilGMSomIyIkWiMiJlMtIkYpKUYmIyIiKkYqRilGKSomIyImXGwiIigrSyU9RikpRiYjIiM2RipGKUYxKiYjIihaWHEiIiorW2cnW0YpKUYmIyIjOEYqRilGKSomIyInTHkiKiIqK2shKVsnRikpRiYjIiM6RipGKUYxKiYjIipuKFtFPSItKz8yRjlKRikpRiYjIiM8RipGKUYpRilJImFHRidGKQ==

print();

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Hence the general solution to the given differential equation will be

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEqU29sbmZpbmFsRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEmc2xsbG5GJ0YvRjItRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZWLUYsNiVRJ3NsbGxuMkYnRi9GMg==

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

print();

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MAPLE dsolve solution

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IiM1

print();

IiM1

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

Ly1JInlHNiI2I0kieEdGJSwmKihJJF9DMUdGJSIiIilGJyNGKyIiI0YrKzlGJ0YrIiIhIyEiJCIiJUYrIyIiKCIjS0YuIyEkTCIiJT8+IiIkIyIkWiMiJlMtIkYzIyEmXGwiIigrSyU9IiImIyIoWlhxIiIqK1tnJ1siIicjISdMeSIqIiorayEpWydGNSMiKm4oW0U9Ii0rPzJGOUoiIikjISxgMS4xdCUiMCtTMW9MZiE+IiIqLUkiT0clKnByb3RlY3RlZEc2I0YrIiM1RitGKyooSSRfQzJHRiVGK0YnRisrOUYnRitGMCMhIiJGOkYrI0YrRlVGLiNGWiIjSUY6IyIjOCIlITMiRjMjISMiKiImKyk+RkEjIiQ8IyInKyk9IkZFIyElTDgiKCs/eSJGNSMiJkZgIyIpK1N5ISlGTCMhJjR0KiIqK2cwRihGUEZRRlVGK0Yr

print();

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<Text-field style="Heading 3" layout="Heading 3">Example 2. Find a series solution for the differential equation <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>about the point x = 0</Text-field>

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x = 0 is a regular singular point for this deq.

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print();

LywoKiYpSSJ4RzYiIiIjIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1GKzYkLUkieUdGJzYjRiZGJkYmRilGKSomRiZGKUYuRikhIiIqJiwmRilGKUYmRjRGKUYwRilGKSIiIQ==

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print();

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

print();

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print();

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

KiYtSSN5MUc2IjYjIiIhIiIiKSwmSSJyR0YlRihGKCEiIiIiI0Yo

print();

KiYtSSN5MUc2IjYjIiIhIiIiKSwmSSJyR0YlRihGKCEiIiIiI0Yo

We see that r = 1 is a double root and proceed to find the first solution using the procedures outlined above.

JSFH

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print();

JSFH

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print();

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PCkvLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiImNiNJI3kxR0YrNiMiIiEsJComIyIiIiIkPyJGNkkiYUdGK0Y2RjYvLS0tRic2JEYtIiIkRi9GMSwkKiYjRjYiIidGNkY4RjZGNi8tLS1GJzYkRi0iIihGL0YxLCQqJiNGNiIlU11GNkY4RjZGNi8tLS1GJzYkRi0iIiNGL0YxLCQqJiNGNkZSRjZGOEY2RjYvLS0tRic2JEYtIiIlRi9GMSwkKiYjRjYiI0NGNkY4RjZGNi8tLS1GJzYkRi0iIilGL0YxLCQqJiNGNiImPy4lRjZGOEY2RjYvLS0tRic2JEYtRkJGL0YxLCQqJiNGNiIkPyhGNkY4RjZGNg==

print();

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

KiYsNEkieEc2IiIiIiokKUYkIiIjRiZGJiomI0YmIiIlRiYpRiQiIiRGJkYmKiYjRiYiI09GJilGJEYsRiZGJiomI0YmIiR3JkYmKUYkIiImRiZGJiomI0YmIiYrVyJGJilGJCIiJ0YmRiYqJiNGJiInKyU9JkYmKUYkIiIoRiZGJiomI0YmIikrO1NERiYpRiQiIilGJkYmKiYjRiYiKytDcUQ7RiYpRiQiIipGJkYmRiZJImFHRiVGJg==

print();

We now proceed to find the second solution using the fact that it will be of the form

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

LCYqJi1JI3kxRzYiNiNJInhHRiYiIiItSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJkYnRilGKS1JJHN1bUdGLDYkKiYmSSJiR0YmNiNJIm5HRiZGKSlGKCwmRjZGKUYpRilGKS9GNjtGKUkpaW5maW5pdHlHRi1GKQ==

print();

LCYqJi1JI3kxRzYiNiNJInhHRiYiIiItSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJkYnRilGKS1JJHN1bUdGLDYkKiYmSSJiR0YmNiNJIm5HRiZGKSlGKCwmRjZGKUYpRilGKS9GNjtGKUkpaW5maW5pdHlHRi1GKQ==

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElZGlmZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJXNsbjJGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInhGJ0YvRjJGQQ==

LCgqJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSN5MUc2IjYjSSJ4R0YqRiwiIiItSSNsbkc2JEYmSShfc3lzbGliR0YqRitGLUYtKiZGKEYtRiwhIiJGLS1JJHN1bUdGMDYkKiomSSJiR0YqNiNJIm5HRipGLSlGLCwmRjtGLUYtRi1GLUY9Ri1GLEYzL0Y7O0YtSSlpbmZpbml0eUdGJkYt

print();

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

LywwKiYsKiomKUkieEc2IiIiIyIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRi02JC1JI3kxR0YpNiNGKEYoRihGK0YrKiZGKEYrRjBGKyEiIkYyRisqJkYoRitGMkYrRjZGKy1JI2xuRzYkRi5JKF9zeXNsaWJHRilGNEYrRisqKEYqRitGKEYrRjBGK0YrKiZGKkYrRjJGK0Y2KiZGJ0YrLUkkc3VtR0Y6NiQsJioqJkkiYkdGKTYjSSJuR0YpRispRihGR0YrKUZHRipGK0YoRjZGKyoqRkRGK0ZIRitGR0YrRihGNkYrL0ZHO0YrSSlpbmZpbml0eUdGLkYrRisqJkYoRistRkA2JCwmKihGREYrRkhGK0ZHRitGKyomRkRGK0ZIRitGK0ZLRitGNiomRihGKy1GQDYkRlNGS0YrRisqJkYoRistRkA2JCooRkRGK0ZIRitGKEYrRktGK0Y2IiIh

print();

Since y1(x) is a solution of the deq the factor on ln(x) is zero (We also convert our series into a finite sum in order to deal with the summations...)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEmc2xuMmFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi9RKiZjb2xvbmVxO0YnRi9GMkY1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYmUShjb2xsZWN0RidGL0YyRjUtSShtZmVuY2VkR0YkNiUtRiM2Ji1GLDYmUSdleHBhbmRGJ0YvRjJGNS1GUzYlLUYjNiQtRiw2JlElc3Vic0YnRi9GMkY1LUZTNiUtRiM2QS1GLDYmUSN5MUYnRi9GMkY1LUZTNiUtRiM2Iy1GLDYmUSJ4RidGL0YyRjVGMi9GNlEnbm9ybWFsRictRjk2L1EiPUYnRi9GMkY1RjxGPkZARkJGREZGRkhGSkZNLUYsNiZRJ3NsbGxuMUYnRi9GMkY1LUY5Ni9RIixGJ0YvRjJGNUY8L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JlEiMkYnRi9GMkY1LUY5Ni9RIn5GJ0YvRjJGNUY8Rj5GQEZCRkRGRkZIRmVwL0ZORmZwRmZvLUY5Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Zpb0Y8Rj5GQEZCRkRGRkZIRmVwRmBxLUZTNiQtRiM2Ji1JJm1mcmFjR0YkNigtRiM2Iy1GOTYtUTAmRGlmZmVyZW50aWFsRDtGJ0Zpb0Y8Rj5GQEZCRkRGRkZIRmVwRmBxLUYjNiUtRiw2I1EhRictRiM2JEZdci1GLDYlRmhvRi9GNUZici8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXnMvJSliZXZlbGxlZEdGNEZhcS1GIzYlLUYsNiVGYW9GL0Y1LUY5Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRmlvL0Y9USZ1bnNldEYnL0Y/Rlt0L0ZBRlt0L0ZDRlt0L0ZFRlt0L0ZHRlt0L0ZJRlt0RmVwRmBxLUZTNiQtRiM2I0ZnckZpb0ZickZpby1GOTYtUSgmbWludXM7RidGaW9GPEY+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZqdC1GanA2JEZccUZpb0ZhcUZlc0ZidC1GOTYtUSIrRidGaW9GPEY+RkBGQkZERkZGSEZpdEZbdS1JJW1zdXBHRiQ2JUZnckZcdS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGYXEtRlM2JC1GIzYmLUkrbXVuZGVyb3ZlckdGJDYnLUY5Ni1RJiZTdW07RidGaW9GPEY+L0ZBRjFGQi9GRUYxL0ZHRjFGSEZlcC9GTlEsMC4xNjY2NjY3ZW1GJy1GIzYlLUYsNiVRIm5GJ0YvRjUtRjk2LUZdcEZpb0Y8Rj5GQEZCRkRGRkZIRkpGTS1GanA2JFEiMUYnRmlvLUYjNiMtRmpwNiRRIzEwRidGaW8vJSdhY2NlbnRHRjQvJSxhY2NlbnR1bmRlckdGNEZici1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EkNS4wRicvJSZkZXB0aEdGXngvJSpsaW5lYnJlYWtHUSVhdXRvRictRlM2JC1GIzYlLUZpcTYoLUYjNictSSVtc3ViR0YkNiUtRiw2JVEiYkYnRi9GNS1GIzYjRmh2LyUvc3Vic2NyaXB0c2hpZnRHRmZ1RmFxLUZidTYlRmdyRmh2RmR1RmFxLUZidTYlRmh2Rlx1RmR1RmR0RmlyRlxzRl9zRmFzRl51LUZpcTYoLUYjNidGX3lGYXFGaXlGYXFGaHZGZHRGaXJGXHNGX3NGYXNGaW9GaW9GZnRGZ3JGYXEtRlM2JC1GIzYmLUZcdjYnRl52RmZ2LUYjNiMtRmpwNiVGZHdGL0Y1RmV3Rmd3RmJyRml3LUZTNiQtRiM2J0ZickZfekZedS1GIzYlRl95RmFxRml5RmJyRmlvRmlvRl51RmdyRmFxLUZTNiQtRiM2KEZlekZickZpd0ZickZfW2xGYnJGaW9GZnRGZ3JGYXEtRlM2JC1GIzYoRmV6RmJyRml3RmJyLUYjNidGX3lGYXFGaXlGYXFGZ3JGYnJGaW9GMkZpb0YyRmlvLUY5Ni5GY3BGMkZpb0Y8RmRwRkBGQkZERkZGSEZlcEZncEZmb0YyRmlvLUY5Ni5GXXBGMkZpb0Y8Rj5GQEZCRkRGRkZIRkpGTS1GanA2JVEiMEYnRjJGaW8=

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

print();

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print();

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print();

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print();

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print();

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print();

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print();

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MAPLE dsolve solution.....

IiM1

print();

IiM1

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Ly1JInlHNiI2I0kieEdGJSwmKihJJF9DMUdGJSIiIkYnRisrOUYnRisiIiFGK0YrI0YrIiIlIiIjI0YrIiNPIiIkI0YrIiR3JkYvI0YrIiYrVyIiIiYjRisiJyslPSYiIicjRisiKSs7U0QiIigjRisiKytDcUQ7IiIpI0YrIi0rVyo9b0oiIiIqLUkiT0clKnByb3RlY3RlZEc2I0YrIiM1RitGKyomSSRfQzJHRiVGKywmKihGJ0YrLUkjbG5HNiRGR0koX3N5c2xpYkdGJUYmRitGLEYrRisqJkYnRisrN0YnISIjRisjISIkRi9GMCMhIzYiJDMiRjMjISNEIiVjTUYvIyEkUCIiJys/VkY4IyEjXCIoK1M9JkY7IyEkQCIiKitTcSNmRj4jISRoKCItK2dMKWZGI0ZBIyElSHIiMCtTJXA9PmY7RkRGRUZJRitGK0YrRis=

print();

JSFH

<Text-field style="Heading 3" layout="Heading 3">Example 3. Find a series solution for the differential equation <Equation executable="false" style="Heading 3" input-equation="" display="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">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</Equation></Text-field>

Note x = 0 is a regular singular point for this deq.

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

LywoKiZJInhHNiIiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYpNiQtSSJ5R0YmNiNGJUYlRiVGJ0YnKiYiIiVGJ0YsRidGJyomRi5GJ0YlRichIiIiIiE=

print();

LywoKiZJInhHNiIiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYpNiQtSSJ5R0YmNiNGJUYlRiVGJ0YnKiYiIiVGJ0YsRidGJyomRi5GJ0YlRichIiIiIiE=

JSFH

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

print();

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

print();

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnZmFjdG9yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmY29lZmZGJ0YvRjItRjY2JC1GIzYmLUYsNiVRJGxoc0YnRi9GMi1GNjYkLUYjNiMtRiw2JVElc2xubkYnRi9GMi9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGSy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZTLyUqc3ltbWV0cmljR0ZTLyUobGFyZ2VvcEdGUy8lLm1vdmFibGVsaW1pdHNHRlMvJSdhY2NlbnRHRlMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGL0YyLUYjNiUtRiw2JVEickYnRi9GMi1GTjYtUSomdW1pbnVzMDtGJ0ZLRlEvRlVGU0ZWRlhGWkZmbkZobi9GW29RLDAuMjIyMjIyMmVtRicvRl5vRmBwLUkjbW5HRiQ2JFEiMUYnRksvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRktGSw==

KihJInJHNiIiIiItSSN5MUdGJDYjIiIhRiUsJiIiJEYlRiNGJUYl

print();

KihJInJHNiIiIiItSSN5MUdGJDYjIiIhRiUsJiIiJEYlRiNGJUYl

roots are 0 and -3 In this case they differ by a positive integer

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

roots are r = 0 and -3. Using the larger root r = 0

JSFH

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

print();

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

PCovLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiImNiNJI3kxR0YrNiMiIiEsJComIyIiIiIiJ0Y2SSJiR0YrRjZGNi8tLS1GJzYkRi0iIiVGL0YxLCQqJiMiIiQiI05GNkkiYUdGK0Y2RjYvLS0tRic2JEYtIiIqRi9GMSwkKiYjRjYiIzpGNkY4RjZGNi8tLS1GJzYkRi0iIihGL0YxLCQqJiNGNiIjNUY2RjhGNkY2Ly0tLUYnNiRGLUY3Ri9GMSwkKiYjRjYiI0BGNkZERjZGNi8tLS1GJzYkRi1GQkYvRjEsJComI0Y2RkJGNkY4RjZGNi8tLS1GJzYkRi0iIilGL0YxLCQqJiNGNiIjTEY2RkRGNkY2Ly0tLUYnNiRGLSIiI0YvRjEsJComI0Y2Ri5GNkZERjZGNg==

print();

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

PCovLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YoIiImNiNJI3kxR0YrNiMiIiFGMi8tLS1GJzYkRi0iIipGL0YxRjIvLS0tRic2JEYtIiIoRi9GMUYyLy0tLUYnNiRGLSIiJEYvRjFGMi8tLS1GJzYkRi0iIiVGL0YxLCQqJiNGRCIjTiIiIkkiYUdGK0ZPRk8vLS0tRic2JEYtIiInRi9GMSwkKiYjRk8iI0BGT0ZQRk9GTy8tLS1GJzYkRi0iIilGL0YxLCQqJiNGTyIjTEZPRlBGT0ZPLy0tLUYnNiRGLSIiI0YvRjEsJComI0ZPRi5GT0ZQRk9GTw==

print();

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

KiYsLCIiIkYkKiYjRiQiIzVGJClJInhHNiIiIiNGJEYkKiYjRiQiJCFHRiQpRikiIiVGJEYkKiYjRiQiJj9eIkYkKUYpIiInRiRGJComI0YkIihnMEwiRiQpRikiIilGJEYkRiRJImFHRipGJA==

print();

KiYsLCIiIkYkKiYjRiQiIzVGJClJInhHNiIiIiNGJEYkKiYjRiQiJCFHRiQpRikiIiVGJEYkKiYjRiQiJj9eIkYkKUYpIiInRiRGJComI0YkIihnMEwiRiQpRikiIilGJEYkRiRJImFHRipGJA==

A second solution of the form 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and where C is a constant that may be zero.

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVElc2xuMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRIkNGJ0YvRjItRjY2LVEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlYtRiw2JVEjeTFGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RidGL0YyRjlGUi1GLDYlUSNsbkYnL0YwRj1GOUZlbi1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmVvLUkrbXVuZGVyb3ZlckdGJDYnLUY2Ni1RJiZTdW07RidGOS9GPFEmdW5zZXRGJy9GP0ZecC9GQUYxL0ZDRl5wL0ZFRjEvRkdGMS9GSUZecEZVL0ZOUSwwLjE2NjY2NjdlbUYnLUYjNiUtRiw2JVEibkYnRi9GMi1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JI21uR0YkNiRRIjBGJ0Y5LUYjNiUtRiw2I1EhRictRiM2Iy1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIRlVGV0ZlcS8lJ2FjY2VudEdGPS8lLGFjY2VudHVuZGVyR0Y9LUYsNiVRImJGJ0YvRjItRmZuNiYtRiM2I0ZpcEY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRlItSSVtc3VwR0YkNiVGam4tRiM2JUZpcC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkhGZG9GZm8tRmBxNiRRIjNGJ0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw==

LCYqKEkiQ0c2IiIiIi1JI3kxR0YlNiNJInhHRiVGJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRilGJkYmLUkkc3VtR0YtNiQqJiZJImJHRiU2I0kibkdGJUYmKUYqLCZGN0YmIiIkISIiRiYvRjc7IiIhSSlpbmZpbml0eUdGLkYm

print();

LCYqKEkiQ0c2IiIiIi1JI3kxR0YlNiNJInhHRiVGJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRilGJkYmLUkkc3VtR0YtNiQqJiZJImJHRiU2I0kibkdGJUYmKUYqLCZGN0YmIiIkISIiRiYvRjc7IiIhSSlpbmZpbml0eUdGLkYm

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmc2xuMjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUShjb2xsZWN0RidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEnZXhwYW5kRidGL0YyLUZTNiQtRiM2JC1GLDYlUSVzdWJzRidGL0YyLUZTNiQtRiM2KC1GLDYlUSJ5RidGL0YyLUZTNiQtRiM2Iy1GLDYlUSJ4RidGL0YyRjktRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVElc2xuMkYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJGRlcUYnRi9GMkY5RjlGX3AtRiw2JVEjbG5GJy9GMEY9RjlGYm9GOQ==

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print();

Again since y1(x) is a solution the factor on ln(x) is 0.

JSFH

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print();

JSFH

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print();

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print();

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LCYqJiwoIiIiRiUqJiNGJSIjNUYlKUkieEc2IiIiI0YlRiUqJiNGJSIkIUdGJSlGKiIiJUYlRiVGJSZJImJHRis2IyIiJEYlRiUqJiwqKiYjRiUiIilGJUYqRiUhIiIqJiNGJSIkVyJGJSlGKkY1RiVGOyokRj9GO0YlKiYjRiVGLEYlRipGO0Y7RiUmRjM2IyIiIUYlRiU=

print();

NOTE The factor on 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 another independent solution we must choose LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2J1EiYkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JC1JI21uR0YkNiZRIjBGJ0YyL0Y4RjxGOi9GOFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRkQ= to be nonzero.

In effect sln2aff is a general solution to the given deq.

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEmT3JkZXJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RKiZjb2xvbmVxO0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIzEwRidGMkY8LUY5Ni5RIjpGJ0YyRjxGPkZARkJGREZGRkhGSkZMRk8=

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

Ly1JInlHNiI2I0kieEdGJSwmKiZJJF9DMUdGJSIiIisvRidGKyIiISNGKyIjNSIiIyNGKyIkIUciIiUjRisiJj9eIiIiJyNGKyIoZzBMIiIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGK0YvRitGKyooSSRfQzJHRiVGKylGJyIiJCEiIisvRiciIzdGLSEiJ0YwIyEiJEYwRjMjRkJGREY2I0ZCIiQhW0Y5RjpGL0YrRis=

print();