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

<Text-field style="Heading 1" layout="Heading 1">Implementing the Classical Algorithm in Maple</Text-field>

Consider the differential equation

de := 2*x*diff(y(x),x,x)+(1+x)*diff(y(x),x)-2*y(x);

LCgqKCIiIyIiIkkieEc2IkYlLUklZGlmZkclKnByb3RlY3RlZEc2JC1GKTYkLUkieUdGJzYjRiZGJkYmRiVGJSomLCZGJUYlRiZGJUYlRixGJUYlKiZGJEYlRi5GJSEiIg==

appearing in Section 107, pages 317-20, in Rainville's Elementary Differential Equations text. At LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk=, Maple provides

dsolve(de, y(x), formal_solution);

Ly1JInlHNiI2I0kieEdGJSwqSSRfQzFHRiUiIiIqKCIiI0YqRilGKkYnRipGKiooI0YqIiIkRipGKUYqKUYnRixGKkYqKipGL0YqKUYnI0YqRixGKkkkX0MyR0YlRiotSSRTdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JCouKSEiIkkkX24xR0YlRiopRiwsJEY+Rj1GKiwmRi9GPSomRixGKkY+RipGKkY9LUkmR0FNTUFHRjc2IywmRj5GKkYqRipGPSwmRipGPSomIiIlRiopRj5GLEYqRipGPSlGJ0Y+RiovRj47IiIhSSlpbmZpbml0eUdGOEYqRio=

as the general solution. Later, we will show that Maple can provide the solution in a simpler form wherein LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkX24xRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn is replaced by the visually simpler LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==.

The classical solution of this equation begins with the determination of its singular points. We obtain these with the Maple command

DEtools[singularities](de);

NiQvSShyZWd1bGFyRzYiPCMiIiEvSSppcnJlZ3VsYXJHRiU8I0kpaW5maW5pdHlHJSpwcm90ZWN0ZWRH

where we will use the long names for commands in the DEtools, gfun, and Slode packages that contain the tools we need for our investigations.

The command just used tells us that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= is a regular singular point, and that the point at infinity is an irregular singular point. All other points are regular points where there are two convergent Taylor series solutions. At the regular singular point, there is at least one solution of the form

Y := Sum(a[k]*x^(k+c),k=0..infinity);

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJiZJImFHRic2I0kia0dGJyIiIilJInhHRicsJkYtRi5JImNHRidGLkYuL0YtOyIiIUkpaW5maW5pdHlHRiU=

The result of substituting this formal solution into the differential equation requires different manipulations on different terms, and this is difficult to do in Maple. It is better to work termwise on the differential equation. For this, we first obtain (and simplify) the first and second derivatives of the formal series.

Y1 := combine(diff(Y,x)); Y2 := combine(diff(Y1,x));

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCZJImFHRic2I0kia0dGJyIiIilJInhHRicsKEYtRi5JImNHRidGLkYuISIiRi4sJkYtRi5GMkYuRi4vRi07IiIhSSlpbmZpbml0eUdGJQ==

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKiwoSSJrR0YnIiIiSSJjR0YnRixGLCEiIkYsJkkiYUdGJzYjRitGLClJInhHRicsKEYrRixGLUYsIiIjRi5GLCwmRitGLEYtRixGLC9GKzsiIiFJKWluZmluaXR5R0Yl

The degree of the generic term in the second derivative has been lowered by 2, and in the first derivative, by 1. When the second derivative is multiplied by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, its generic term will be comparable to that in the first derivative. Hence, we form and simplify the sum

T1 := factor(combine(2*x*Y2 + Y1));

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKiZJImFHRic2I0kia0dGJyIiIilJInhHRicsKEYtRi5JImNHRidGLkYuISIiRi4sJkYtRi5GMkYuRi4sKComIiIjRi5GMkYuRi4qJkY3Ri5GLUYuRi5GLkYzRi4vRi07IiIhSSlpbmZpbml0eUdGJQ==

When the first derivative is multiplied by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, the generic term will be comparable with that in 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 itself, so we form and simplify the sum

T2 := factor(combine(x*Y1-2*Y));

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCZJImFHRic2I0kia0dGJyIiIilJInhHRicsJkYtRi5JImNHRidGLkYuLChGLUYuRjJGLiIiIyEiIkYuL0YtOyIiIUkpaW5maW5pdHlHRiU=

To combine these to formal sums, we need to "lower" the index in the second sum. We do this in Maple with

temp := dchange(k=n-1,T2): T3 := dchange(n=k,temp);

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKCZJImFHRic2IywmSSJrR0YnIiIiRi8hIiJGLylJInhHRicsKEYuRi9JImNHRidGL0YvRjBGLywoRi5GLyIiJEYwRjRGL0YvL0YuO0YvSSlpbmZpbml0eUdGJQ==

Before we can add the sum starting at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= with the new sum that starts with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JEZURjk=, we need to extract the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= term from that first sum. It turns out best to extract the summand with

S1 := Summand(T1)[1];

KiomSSJhRzYiNiNJImtHRiUiIiIpSSJ4R0YlLChGJ0YoSSJjR0YlRihGKCEiIkYoLCZGJ0YoRixGKEYoLCgqJiIiI0YoRixGKEYoKiZGMUYoRidGKEYoRihGLUYo

evaluate this summand at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= to get

q := eval(S1,k=0);

KiomSSJhRzYiNiMiIiEiIiIpSSJ4R0YlLCZGKCEiIkkiY0dGJUYoRihGLUYoLCYqJiIiI0YoRi1GKEYoRihGLEYo

then rebuild the resulting sum as

T4 := Sum(S1,k=1..infinity);

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKiZJImFHRic2I0kia0dGJyIiIilJInhHRicsKEYtRi5JImNHRidGLkYuISIiRi4sJkYtRi5GMkYuRi4sKComIiIjRi5GMkYuRi4qJkY3Ri5GLUYuRi5GLkYzRi4vRi07Ri5JKWluZmluaXR5R0Yl

The extracted LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= term leads to the indicial equation

IndicialEquation := remove(has,q,{x,a}) = 0;

LyomSSJjRzYiIiIiLCYqJiIiI0YmRiRGJkYmRiYhIiJGJiIiIQ==

whose roots are

solve(IndicialEquation,c);

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Thus, at the regular singular point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk=, one solution will be a Taylor series, and the other will be a Taylor series multiplied by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkmbXNxcnRHRiQ2Iy1GLDYlUSJ4RidGL0YyRis=.

To determine the general coefficient in either of these series solutions, we add and simplify the two sums that now have comparable generic terms and identical starting indices. We therefore obtain

T5 := factor(combine(T4+T3));

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJilJInhHRicsKEkia0dGJyIiIkkiY0dGJ0YuRi4hIiJGLiwyKioiIiVGLiZJImFHRic2I0YtRi5GLUYuRi9GLkYuKigiIiNGLkY0Ri4pRi1GOEYuRi4qJkY0Ri5GLUYuRjAqKEY4Ri5GNEYuKUYvRjhGLkYuKiZGNEYuRi9GLkYwKiYmRjU2IywmRi1GLkYuRjBGLkYtRi5GLiomIiIkRi5GP0YuRjAqJkY/Ri5GL0YuRi5GLi9GLTtGLkkpaW5maW5pdHlHRiU=

The identical vanishing of each coefficient is assured by equating the generic coefficient in this sum to zero. Isolating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRImFGJ0YvRjItRiw2JVEia0YnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= leads to the recursion equation

RecursionEqn := factor(isolate(Summand(T5)[1]=0,a[k]));

LyZJImFHNiI2I0kia0dGJSwkKiomRiQ2IywmRiciIiJGLSEiIkYtLChGJ0YtIiIkRi5JImNHRiVGLUYtLCZGJ0YtRjFGLUYuLCgqJiIiI0YtRjFGLUYtKiZGNUYtRidGLUYtRi1GLkYuRi4=

Without some of the software tools now available, the student solving this recurrence equation has to consider the two separate recursions corresponding to the two indicial roots, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= and 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. For each, the recursion is used to generate a sequence of terms from which a general pattern must be deduced. Instead, we will explore the use of Maple's rsolve command, the solver designed for recursion equations.

Unfortunately, this solver uses function notation, not subscript notation, for the unknown. Hence, we write

Q := eval(RecursionEqn, [a[k]=u(k), a[k-1]=u(k-1)]);

Ly1JInVHNiI2I0kia0dGJSwkKiotRiQ2IywmRiciIiJGLSEiIkYtLChGJ0YtIiIkRi5JImNHRiVGLUYtLCZGJ0YtRjFGLUYuLCgqJiIiI0YtRjFGLUYtKiZGNUYtRidGLUYtRi1GLkYuRi4=

Evaluating this equation for each indicial root gives the recursions

Q0 := eval(Q, c=0); Q1 := simplify(eval(Q, c=1/2));

Ly1JInVHNiI2I0kia0dGJSwkKiotRiQ2IywmRiciIiJGLSEiIkYtLCZGJ0YtIiIkRi5GLUYnRi4sJkYtRi4qJiIiI0YtRidGLUYtRi5GLg==

Ly1JInVHNiI2I0kia0dGJSwkKiwjIiIiIiIjRistRiQ2IywmRidGK0YrISIiRissJiomRixGK0YnRitGKyIiJkYwRissJkYyRitGK0YrRjBGJ0YwRjA=

whose solutions are, respectively,

U0 := rsolve(Q0, u(k)): U1 := rsolve(Q1, u(k)): u(k) = U0; u(k) = U1;

Ly1JInVHNiI2I0kia0dGJS1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JTMxIiIhRicyRiciIiQsJCowIiIjIiIiKSEiIkYnRjQpRjMsJEYnRjZGNC1JJkdBTU1BRzYkRipJKF9zeXNsaWJHRiU2IywmRidGNCNGNEYzRjRGNilJI1BpR0YqRj9GNCwoKiZGMEY0KUYnRjNGNEY0KiYiIidGNEYnRjRGNkYzRjRGNi1GJDYjRi5GNEY0Ri4=

Ly1JInVHNiI2I0kia0dGJSwkKjAiIiQiIiIpISIiRidGKykiIiMsJEYnRi1GKywmKiZGL0YrRidGK0YrRipGLUYtLCZGK0YtKiYiIiVGKylGJ0YvRitGK0YtLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmRidGK0YrRitGLS1GJDYjIiIhRitGKw==

Careful inspection of the first result shows that 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 This is clear from

eval(U0, k=1); eval(U0, k=2); eval(U0, k=3);

LCQqJiIiIyIiIi1JInVHNiI2IyIiIUYlRiU=

LCQqJiMiIiIiIiRGJS1JInVHNiI2IyIiIUYlRiU=

IiIh

The second result generates the coefficients

for n from 0 to 5 do eval(U1, k=n); end do;

LUkidUc2IjYjIiIh

LCQqJiMiIiIiIiNGJS1JInVHNiI2IyIiIUYlRiU=

LCQqJiMiIiIiI1NGJS1JInVHNiI2IyIiIUYlRiU=

LCQqJiMiIiIiJSFvIkYlLUkidUc2IjYjIiIhRiUhIiI=

LCQqJiMiIiIiJj8uJUYlLUkidUc2IjYjIiIhRiVGJQ==

LCQqJiMiIiIiJ1NxKSlGJS1JInVHNiI2IyIiIUYlISIi

whose pattern is not immediately obvious. Table 1 contains the first few individual relations generated by the recursion.

for n from 1 to 6 do eval(Q1, k=n); end do;

Ly1JInVHNiI2IyIiIiwkKiYjRiciIiNGJy1GJDYjIiIhRidGJw==

Ly1JInVHNiI2IyIiIywkKiYjIiIiIiM/RistRiQ2I0YrRitGKw==

Ly1JInVHNiI2IyIiJCwkKiYjIiIiIiNVRistRiQ2IyIiI0YrISIi

Ly1JInVHNiI2IyIiJSwkKiYjIiIiIiNDRistRiQ2IyIiJEYrISIi

Ly1JInVHNiI2IyIiJiwkKiYjIiIiIiNBRistRiQ2IyIiJUYrISIi

Ly1JInVHNiI2IyIiJywkKiYjIiIoIiRjIiIiIi1GJDYjIiImRi0hIiI=

Table 1 Recursions for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRGVEY5LUZmbjYkUSIyRidGOS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGPS1GLDYlUSFGJ0YvRjI=

Table 1 shows that 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 is given in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVEidUYnRi9GMi1JI21vR0YkNjBRMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZULyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkobWZlbmNlZEdGJDYkLUYjNiMtSSNtbkdGJDYkUSIwRidGPkY+Ris= as a product of the factors of the 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 appearing on the right side of each of the equations exemplified in the table. Hence, 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 would be given by the product of the factors

P := coeff(rhs(Q1), u(k-1));

LCQqKiMiIiIiIiNGJSwmKiZGJkYlSSJrRzYiRiVGJSIiJiEiIkYlLCZGKEYlRiVGJUYsRilGLEYs

that is, by

unassign('n'); un := value(Product(P,k=1..n)): u(n) = un;

Ly1JInVHNiI2I0kibkdGJSwkKiwjIiIkIiIlIiIiKSMhIiIiIiMsJkYnRi1GLUYtRi0tSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCZGJ0YtI0YrRjFGMEYtLUY0NiMsJkYnRi1GOkYtRjAtRjQ2I0YyRjBGMA==

Thus, we have determined that the general solution of the differential equation is a linear combination of a finite polynomial (in fact, a quadratic) and a Taylor series multiplied by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkmbXNxcnRHRiQ2Iy1GLDYlUSJ4RidGL0YyRis=.

A complete series solution of even a second-order linear differential equation clearly takes a considerable amount of manipulative skill if the solution is to be generated by hand using the classical algorithms. Let us next see how Maple can provide the full solution with the least effort on the part of the user.

<Text-field style="Heading 1" layout="Heading 1">Maple's Minimum-Effort Solution</Text-field>

Given the differential equation

de;

LCgqKCIiIyIiIkkieEc2IkYlLUklZGlmZkclKnByb3RlY3RlZEc2JC1GKTYkLUkieUdGJzYjRiZGJkYmRiVGJSomLCZGJUYlRiZGJUYlRixGJUYlKiZGJEYlRi5GJSEiIg==

we showed how Maple provides the singular points via

DEtools[singularities](de);

NiQvSShyZWd1bGFyRzYiPCMiIiEvSSppcnJlZ3VsYXJHRiU8I0kpaW5maW5pdHlHJSpwcm90ZWN0ZWRH

At the regular singular point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk=, the differential equation has a fundamental set consisting of at least one series of the form

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJiZJImFHRic2I0kia0dGJyIiIilJInhHRicsJkYtRi5JImNHRidGLkYuL0YtOyIiIUkpaW5maW5pdHlHRiU=

To determine the appropriate values of the parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, we have

DEtools[indicialeq](de, x, 0, y(x));

LywmKiQpSSJ4RzYiIiIjIiIiRikqJiNGKUYoRilGJkYpISIiIiIh

the automatic generation of the indicial equation whose roots are clearly LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= and 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.

Maple further classifies the points as hypergeometric, rational, or polynomial (respectively) with the command

Slode[candidate_points](de, y(x));

NyU8JCIiIUkzYW55X29yZGluYXJ5X3BvaW50RzYiPCQhIiJGJDwjRig=

Hypergeometric points are those about which the coefficients in the series solutions obey linear recursion equations. Consequently, at the regular singular point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk=, a fundamental set of solutions for the differential equation can be generated by the single Maple command

FS := Slode[hypergeom_formal_sol](de, y(x));

NyQsKCIiIkYkKiYiIiNGJEkieEc2IkYkRiQqJiNGJCIiJEYkKUYnRiZGJEYkLCQqKEYrRiQpRicjRiRGJkYkLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiQqMCkhIiJJIm5HRihGJClGJiwkRjpGOUYkLUkmR0FNTUFHRjM2IywmRjpGJEYkRiRGOSwmKiZGJkYkRjpGJEYkRiRGJEY5LCZGQkYkRiRGOUY5LCZGK0Y5RkJGJEY5KUYnRjpGJC9GOjsiIiFJKWluZmluaXR5R0Y0RiRGJA==

Thus, we have recovered all the information of the previous section, except for the recurrence equations satisfied by the coefficients of the series solutions. Before we see how Maple might provide us these recursions, we first make sure that the series solution given here agrees with the results from the previous section were we found the general coefficient to be

un;

LCQqLCMiIiQiIiUiIiIpIyEiIiIiIywmSSJuRzYiRidGJ0YnRictSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGLjYjLCZGLUYnI0YlRitGKkYnLUYwNiMsJkYtRidGNkYnRiotRjA2I0YsRipGKg==

(The polynomial solutions are clearly the same.) To begin a comparison of the two series solutions, we first extract the new solution via

FS1 := select(has,FS,GAMMA)[1];

LCQqKCIiJCIiIilJInhHNiIjRiUiIiNGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYkKjApISIiSSJuR0YoRiUpRiosJEY0RjNGJS1JJkdBTU1BR0YtNiMsJkY0RiVGJUYlRjMsJiomRipGJUY0RiVGJUYlRiVGMywmRjxGJUYlRjNGMywmRiRGM0Y8RiVGMylGJ0Y0RiUvRjQ7IiIhSSlpbmZpbml0eUdGLkYlRiU=

then select the summand with

FS2 := Summand(FS1)[1];

KjApISIiSSJuRzYiIiIiKSIiIywkRiVGJEYnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2IywmRiVGJ0YnRidGJCwmKiZGKUYnRiVGJ0YnRidGJ0YkLCZGM0YnRidGJEYkLCYiIiRGJEYzRidGJClJInhHRiZGJUYn

Unfortunately, this expression contains "local" variables that we modify with

FS3 := convert(FS2,`global`);

KjApISIiSSJuRzYiIiIiKSIiIywkRiVGJEYnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2IywmRiVGJ0YnRidGJCwmKiZGKUYnRiVGJ0YnRidGJ0YkLCZGM0YnRidGJEYkLCYiIiRGJEYzRidGJClJInhHRiZGJUYn

Since we want to compare the expressions for the coefficients of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRInhGJ0YvRjItRiw2JVEibkYnRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKw== in the two versions of the series we generated, write

FS4 := remove(has,FS3,x^n);

Ki4pISIiSSJuRzYiIiIiKSIiIywkRiVGJEYnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2IywmRiVGJ0YnRidGJCwmKiZGKUYnRiVGJ0YnRidGJ0YkLCZGM0YnRidGJEYkLCYiIiRGJEYzRidGJA==

and compare via the ratio

simplify(FS4/un);

IyIiIiIiJA==

Thus, the two versions of the series solution differ by just a constant multiple, so we conclude that both series solutions are essentially the same.

The final item the classical algorithm provides is the recurrence relation satisfied by the coefficients in the series solutions. The Maple commands

R1 := gfun[diffeqtorec](de, y(x), u(n)); R2 := Slode[FPseries](de, y(x), u(n));

LCYqJiwmIiIjISIiSSJuRzYiIiIiRiktSSJ1R0YoNiNGJ0YpRikqJiwoKiYiIiRGKUYnRilGKUYpRikqJkYlRikpRidGJUYpRilGKS1GKzYjLCZGJ0YpRilGKUYpRik=

LUkqRlBTc3RydWN0RzYiNiQsJiZJI19DR0YkNiMiIiEiIiItSSRTdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JComLUkidUdGJDYjSSJuR0YkRispSSJ4R0YkRjZGKy9GNjtGK0kpaW5maW5pdHlHRi9GKywmKiYsJkY2ISIiKiYiIiNGKylGNkZBRitGK0YrRjNGK0YrKiYsJiIiJEY/RjZGK0YrLUY0NiMsJkY2RitGK0Y/RitGKw==

both provide just one recursion relation each. Unfortunately, both are for the series with indicial root LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk=, as we see from the solutions

rsolve(R1, u(n));

LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYlMzEiIiFJIm5HNiIyRikiIiQsJCowIiIjIiIiKSEiIkYpRjApRi8sJEYpRjJGMC1JJkdBTU1BRzYkRiRJKF9zeXNsaWJHRio2IywmRilGMCNGMEYvRjBGMilJI1BpR0YkRjtGMCwoKiYiIidGMEYpRjBGMiomRixGMClGKUYvRjBGMEYvRjBGMi1JInVHRio2I0YoRjBGMEYo

and

rsolve(op(remove(has,R2,x)), u(n));

Obtaining the recursion equation for the indicial root 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 will be considered next.

<Text-field style="Heading 1" layout="Heading 1">Truncated Series</Text-field>

In a variety of ways, Maple can provide the solution of a differential equation as a truncated series. First, we have

dsolve(de, y(x), series);

Ly1JInlHNiI2I0kieEdGJSwmKihJJF9DMUdGJSIiIilGJyNGKyIiI0YrKzFGJ0YrIiIhRi1GKyNGKyIjU0YuIyEiIiIlIW8iIiIkI0YrIiY/LiUiIiUjRjQiJ1NxKSkiIiYtSSJPRyUqcHJvdGVjdGVkRzYjRisiIidGK0YrKiZJJF9DMkdGJUYrKytGJ0YrRjBGLkYrI0YrRjZGLkY9RkFGK0Yr

In this form, a truncated expansion of the general solution, it's not clear that the term without the factor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkmbXNxcnRHRiQ2Iy1GLDYlUSJ4RidGL0YyRis= is actually a polynomial. Alternatively, we have

DEtools[formal_sol](de, y(x));

NyQrK0kieEc2IiIiIiIiISIiI0YmI0YmIiIkRigtSSJPRyUqcHJvdGVjdGVkRzYjRiYiIicqJilGJCNGJkYoRiYrMUYkRiZGJ0YyRiYjRiYiI1NGKCMhIiIiJSFvIkYqI0YmIiY/LiUiIiUjRjciJ1NxKSkiIiZGK0YvRiY=

with the general solution replaced by approximations to the members of the fundamental set.

Given either recursion

Q0; Q1;

Ly1JInVHNiI2I0kia0dGJSwkKiotRiQ2IywmRiciIiJGLSEiIkYtLCZGJ0YtIiIkRi5GLUYnRi4sJkYtRi4qJiIiI0YtRidGLUYtRi5GLg==

Ly1JInVHNiI2I0kia0dGJSwkKiwjIiIiIiIjRistRiQ2IywmRidGK0YrISIiRissJiomRixGK0YnRitGKyIiJkYwRissJkYyRitGK0YrRjBGJ0YwRjA=

the command rectoproc writes a procedure whose values are members of the sequence satisfying the recursion. Thus, we have the procedures

C0 := gfun[rectoproc](Q0, u(k)): C1 := gfun[rectoproc](Q1, u(k)):

that generate the respective members

seq(C0(k), k=0..6); seq(C1(k), k=0..6);

NikmSSNfQ0c2IjYjIiIhLCQqJiIiIyIiIkYjRitGKywkKiYjRisiIiRGK0YjRitGK0YnRidGJ0Yn

NikmSSNfQ0c2IjYjIiIhLCQqJiMiIiIiIiNGK0YjRitGKywkKiYjRisiI1NGK0YjRitGKywkKiYjRisiJSFvIkYrRiNGKyEiIiwkKiYjRisiJj8uJUYrRiNGK0YrLCQqJiNGKyInU3EpKUYrRiNGK0Y1LCQqJiNGKyIpPyRvKD5GK0YjRitGKw==

Notice that in each case, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRistSSVtc3ViR0YkNiUtRiw2JVEjX0NGJ0YvRjItSSNtbkdGJDYkUSIwRicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKw== is used as the initial value in the sequence.

In the event that a command such as

dsolve(de, y(x), formal_series);

NiQvLUkieUc2IjYjSSJ4R0YmLCgqJkkkX0MxR0YmIiIiLUkqZmFjdG9yaWFsRyUqcHJvdGVjdGVkRzYjIiIhISIiRiwqKiIiI0YsRitGLCwmRixGLEYoRixGLC1GLjYjRixGMkYyKihGK0YsKUY1RjRGLC1GLjYjRjRGMkYyL0YkLChGK0YsKihGNEYsRitGLEYoRixGLCooI0YsIiIkRixGK0YsKUYoRjRGLEYs

does not produce two series about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= the way that

Slode[hypergeom_formal_sol](de, y(x));

NyQsJCooIiIkIiIiKUkieEc2IiNGJiIiI0YmLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiQqMCkhIiJJIm5HRilGJilGKywkRjVGNEYmLUkmR0FNTUFHRi42IywmRjVGJkYmRiZGNCwmRiZGJiomRitGJkY1RiZGJkY0LCZGPUYmRiZGNEY0LCZGJUY0Rj1GJkY0KUYoRjVGJi9GNTsiIiFJKWluZmluaXR5R0YvRiZGJiwoRiZGJiomRitGJkYoRiZGJiomI0YmRiVGJilGKEYrRiZGJg==

does, the Maple Research Fellow Dr. Edgardo Cheb-Terab suggests using

dsolve(de, y(x), series);

to determine that there is a solution of the form 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. To find 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, make the change of variables 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 via

Temp := PDEtools[dchange](y(x)=sqrt(x)*g(x), de, [g(x)]);

LCgqKCIiIyIiIkkieEc2IkYlLCgqKCNGJSIiJUYlKUYmIyIiJEYkISIiLUkiZ0dGJzYjRiZGJUYvKiYpRiYjRiVGJEYvLUklZGlmZkclKnByb3RlY3RlZEc2JEYwRiZGJUYlKiZGNEYlLUY3NiRGNkYmRiVGJUYlRiUqJiwmRiVGJUYmRiVGJSwmKihGNUYlRjRGL0YwRiVGJSomRjRGJUY2RiVGJUYlRiUqKEYkRiVGNEYlRjBGJUYv

and extract

DE := remove(has,simplify(Temp),sqrt(x));

LCoqJiIiJCIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJnRzYiNiNJInhHRixGLkYlRiUqKCIiI0YlRi5GJS1GJzYkRiZGLkYlRiUqJiNGJEYwRiVGKkYlISIiKiZGLkYlRiZGJUYl

from which we get the recursion

Qg := gfun[diffeqtorec](DE, g(x), v(k));

LCYqJiwmKiYiIiMiIiJJImtHNiJGJ0YnIiIkISIiRictSSJ2R0YpNiNGKEYnRicqJiwoKiYiIzVGJ0YoRidGJyIiJ0YnKiYiIiVGJylGKEYmRidGJ0YnLUYtNiMsJkYoRidGJ0YnRidGJw==

and its solution

rsolve(Qg, v(k));

LCQqMCIiJCIiIikhIiJJImtHNiJGJSkiIiMsJEYoRidGJSwmKiZGK0YlRihGJUYlRiRGJ0YnLCZGJUYnKiYiIiVGJSlGKEYrRiVGJUYnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRik2IywmRihGJUYlRiVGJy1JInZHRik2IyIiIUYlRiU=

which compares favorably with

U1;

LCQqMCIiJCIiIikhIiJJImtHNiJGJSkiIiMsJEYoRidGJSwmKiZGK0YlRihGJUYlRiRGJ0YnLCZGJUYnKiYiIiVGJSlGKEYrRiVGJUYnLUkmR0FNTUFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRik2IywmRihGJUYlRiVGJy1JInVHRik2IyIiIUYlRiU=

the recursion found earlier for the indicial root 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.