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Tips and Techniques:Eigenvalue Problems for ODEsRobert J. LopezEmeritus Professor of Mathematics and Maple FellowMaplesoftJSFH<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>JSFHSome boundary value problems for partial differential equations are amenable to analytic techniques. For example, the constant-coefficient, second-order linear equations called the heat, wave, and potential equations are solved with some type of Fourier series representation obtained from the Sturm-Liouville eigenvalue problem that arises upon separating variables.The role of Maple in the solution of such boundary value problems is examined. Efficient techniques for separating variables, and a way to guide Maple through the solution of the resulting Sturm-Liouville eigenvalue problems are shown. This is the second edition of a May 2007 article originally written with Maple 11. It necessarily made ample use of Maple commands. This present version presents the same material, only in a syntax-free mode. The only exception is the use of the pdsolve command, for which no Context Menu access is yet available.<Text-field style="Heading 1" layout="Heading 1">Initializations</Text-field>
To gain some notational enhancements, select "Typesetting Rules..." from the View menu, and adjust the settings in the lower-left portion to match those shown in Figure 1. In the space to the left of the "Add Function" button, enter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= and press that button. Then enter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= and again press the "Add Function" button. (The drop-down to the left of the "Remove" button allows for removing dependency suppression from a function.) Finally, press the "Done" button in the bottom-right portion of the Assistant.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1">Separating Variables in a PDE</Text-field>Maple's pdsolve command provides, as an eigenfunction expansion, the solution to the following BVP for the one-dimensional wave equation.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JSFH
JSFHOf course, the eigenfunctions are obtained by separating variables (assume 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) and solving the Sturm-Liouville BVP problem 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 that results. Table 1 summarizes the Maple manipulations that will separate variables in the one-dimensional wave equation.
The wave equationCalculus palette: Partial differential operatorsContext Menu: Assign to a Name\342\211\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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMWFzc2lnbn50b35hfm5hbWVGJ0YvRjJGNUZFPDE := : 'PDE';Define a separated solutionContext Menu: Assign to a Name\342\211\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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMWFzc2lnbn50b35hfm5hbWVGJ0YvRjJGNUZFU := : 'U';Separate variablesExpression palette: Evaluation template
Press the Enter key.Divide by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiZRIlVGJ0YyRjVGOEY1RkE= and press the Enter key.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Extract the two ODEs formed by introducing the Bernoulli separation constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0Reference by its equation label;
Press the Enter key.Context Menu: Left-hand SideEquate to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 and multiply by 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.Context Menu: Move to Left Reference by its equation label;
Press the Enter key.Context Menu: Right-hand SideEquate to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 and multiply by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn.Context Menu: Move to Left 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lhs( );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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRLW1vdmV+dG9+bGVmdEYnRi9GMkY1RkU=lhs() - rhs() = 0;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1Fldy8qKClJImNHNiIiIiMhIiItSSJURzYiNiNJInRHNiIhIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIlRHNiI2I0kidEc2IkkidEc2IkkidEc2IiIiIiomLUkiWEc2IjYjSSJ4RzYiISIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJYRzYiNiNJInhHNiJJInhHNiJJInhHNiIiIiJGJy8lK2FjdGlvbnR5cGVHUTptYXBsZXNvZnQuY29tOmxhYmVsKEwzNTEpRicvJSV2aWV3R1EmbGFiZWxGJ0YrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMHJpZ2h0fmhhbmR+c2lkZUYnRi9GMkY1RkU=rhs( );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lhs() - rhs() = 0;Table 1 Separating variables in the one-dimensional wave equation
JSFHThe boundary conditions 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 yield the homogeneous Dirichlet conditions 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, which are separated.<Text-field style="Heading 1" layout="Heading 1">Solving the Sturm-Liouville Eigenvalue Problem</Text-field>JSFHMaple's dsolve command (invoked through the Context Menu), cannot solve even a simple a Sturm-Liouville eigenvalue problem:JSFH
JSFHJSFHHowever, with a little adroit intervention, this shortcoming can be overcome. First, ascertain any value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 for which the characteristic equation has a multiple root. Find the general solution of that solution separately from the general solution in the case where the characteristic roots are distinct.Apply the boundary conditions to both of these general solutions. Write the resulting (homogeneous) equations in matrix form, and set the determinant equal to zero. Solve the resulting equation for the eigenvalues, and use these eigenvalues to specify the associated eigenfunctions. The technique is illustrated in the following two examples.<Text-field style="Heading 2" layout="Heading 2">Homogeneous Dirichlet Conditions</Text-field>JSFHMake the "exponential guess" LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JJW1zdXBHRiQ2JS1GLDYmUSJlRidGL0YyRjUtRiM2KC1GLDYmUSJtRidGL0YyRjUtRjk2LlEifkYnRjJGPEY+RkBGQkZERkZGSEZKL0ZNUSYwLjBlbUYnL0ZQRmpuLUYsNiZRInhGJ0YvRjJGNUYvRjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMkY8 to obtain and solve the characteristic equation.JSFH
Expression palette: Evaluation template
Press the Enter key.Context Menu: Solve\342\211\273Obtain Solutions for\342\211\273mLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkmbWZyYWNHRiQ2KC1GIzYoRistSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1Fgci8sJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiWEc2IjYjSSJ4RzYiSSJ4RzYiSSJ4RzYiIiIiKiYtSSJYRzYiNiNJInhHNiIiIiJJJ2xhbWJkYUc2IiIiIiEiIiIiIUYnLyUrYWN0aW9udHlwZUdROm1hcGxlc29mdC5jb206bGFiZWwoTDM4MilGJy8lJXZpZXdHUSdvdXRwdXRGJ0YrLyUscGxhY2Vob2xkZXJHUSV0cnVlRicvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRkUvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GIzYmRistRiM2JkYrLUYjNidGKy1GLDYlUSJ4RicvJSdpdGFsaWNHRkUvRklRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnRkgvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRltvLyUpc3RyZXRjaHlHRltvLyUqc3ltbWV0cmljR0Zbby8lKGxhcmdlb3BHRltvLyUubW92YWJsZWxpbWl0c0dGW28vJSdhY2NlbnRHRltvLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGam8tRiw2JVEiYUYnRldGWUZIRitGSEYrRkgvJSxjb25zdHJhaW50c0dRLGhlaWdodC1vbmx5RicvJS5saW5ldGhpY2tuZXNzR1EiMEYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmhwLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUZmbjYtUSkmdmVyYmFyO0YnRkhGaW5GXG8vRl9vRkVGYG9GYm9GZG9GZm8vRmlvUSwwLjExMTExMTFlbUYnL0ZccEZjcS1GMjYoLUZMNiQtRiM2JkYrLUYjNiVGKy1GLDYlUSVmKHgpRidGV0ZZRkhGK0ZIRmBwLUYjNictRiw2JVEiWEYnRldGWS1GZm42LUZobkZIL0ZqbkZdcS9GXW9GXXEvRl9vRl1xL0Zhb0ZdcS9GY29GXXEvRmVvRl1xL0Znb0ZdcUZob0ZbcC1GIzYlRistSSVtc3VwR0YkNiUtRmZuNi1RLyZFeHBvbmVudGlhbEU7RidGSEZpbkZcb0Zeb0Zgb0Zib0Zkb0Zmby9GaW9RJjAuMGVtRicvRlxwRmdzLUYjNictRiw2JVEibUYnRldGWS1GZm42LVEifkYnRkhGZ3JGaHJGaXJGanJGW3NGXHNGXXNGZnNGaHNGVEZXRlkvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZlcEZIRitGSEZjcEZmcEZpcEZbcUZIRisvJStleGVjdXRhYmxlR0ZdcUZILUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMHNvbHV0aW9uc35mb3J+bUYnRi9GMkY1RkU=solve( , m );
JSFHThe characteristic equation 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 shows that for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjBGJ0YyRjRGMkY0, the ODE 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 has the double root LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8, so the general solution in this case is a polynomial.
Expression palette: Evaluation template
Press the Enter key.Context Menu: Solve DE\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=Context Menu: Right-hand 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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRKXNvbHZlfkRFRidGL0YyRjVGRQ==dsolve( , { X(x) } );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMHJpZ2h0fmhhbmR+c2lkZUYnRi9GMkY1RkU=rhs( );JSFH
JSFHAside: Right after obtaining , it would seem natural to invoke the Context Menu option "Assign Function", thereby obtaining the solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= as a function. Then, the boundary conditions 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 would most easily be obtained. Unfortunately, this would render the differential equation opaque to the Context Menu system thereafter. Since is still needed several times in the sequel, the additional step allows the solution to be given the different name, S.
Context Menu: Assign 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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRM2Fzc2lnbn5hc35mdW5jdGlvbkYnRi9GMkY1RkU=proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );Apply the boundary conditions to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA=;
Press the Enter key.Context Menu: Solve\342\211\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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRJnNvbHZlRidGL0YyRjVGRQ==solve( { } );
Clearly, this implies LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RIj1GJ0YyRkAvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWLUkjbW5HRiQ2JVEiMEYnRjJGQEYyRkA=, so LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjBGJ0YyRjRGMkY0 is not an eigenvalue. The general case where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSsmTm90RXF1YWw7RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjBGJ0YyRjRGMkY0 proceeds along similar lines, except for the algebra of solving for the eigenvalues. Begin by unassigning S.
Unassign the name SContext Menu: UnassignLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRKXVuYXNzaWduRidGL0YyRjVGRQ==unassign(''); ;Obtain the general solution of the ODE as the function SReference the ODE by equation label;
Press the Enter key.Context Menu: Solve DEContext Menu: Right-hand SideWrite 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 (using equation label)
Context Menu: Assign FunctionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1Fgci8sJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiWEc2IjYjSSJ4RzYiSSJ4RzYiSSJ4RzYiIiIiKiYtSSJYRzYiNiNJInhHNiIiIiJJJ2xhbWJkYUc2IiIiIiEiIiIiIUYnLyUrYWN0aW9udHlwZUdROm1hcGxlc29mdC5jb206bGFiZWwoTDM4MilGJy8lJXZpZXdHUSZsYWJlbEYnRisvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRKXNvbHZlfkRFRidGL0YyRjVGRQ==dsolve( , { X(x) } );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMHJpZ2h0fmhhbmR+c2lkZUYnRi9GMkY1RkU=rhs( );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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRM2Fzc2lnbn5hc35mdW5jdGlvbkYnRi9GMkY1RkU=proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );
At this point, access to the tools of linear algebra becomes necessary.
Tools\342\211\273Load Package: Student Linear AlgebraLoading Student:-LinearAlgebrawith(Student:-LinearAlgebra):Apply the boundary conditions and solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0Apply the boundary conditions and press the Enter key.Context Menu: Student Linear Algebra\342\211\273Constructions\342\211\273Generate Matrix\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EkX0MxRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiLEYnRjJGNS9GOVEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYnUSRfQzJGJ0YvRjJGNUY4RjJGPw==Context Menu: Select Element\342\211\2731Context Menu: Student Linear Algebra\342\211\273Standard Operations\342\211\273DeterminantContext Menu: Solve\342\211\273Solve (general solution)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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRL3Rvfk1hdHJpeH5mb3JtRidGL0YyRjVGRQ==Student:-LinearAlgebra:-GenerateMatrix( [ ], [_C1,_C2 ], augmented=false );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRL3NlbGVjdH5lbnRyeX4xRidGL0YyRjVGRQ==()[1];LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRLGRldGVybWluYW50RidGL0YyRjVGRQ==Student:-LinearAlgebra:-Determinant( );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRJnNvbHZlRidGL0YyRjVGRQ==solve( { = 0 }, AllSolutions=true );
The symbol LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEkX1oxRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= represents an arbitrary integer, which henceforth will be called n. It remains to see what happens to the general solution of the ODE when 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, for integer n. It suffices to take n as positive.JSFH
JSFHIs all of the eigenfunction? In fact, does all of even satisfy the boundary conditions? JSFH
Apply the boundary conditions to Expression palette: Evaluation template
Press the Enter key.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Expression palette: Evaluation template
(Include result from first condition.)
Press the Enter key.Context Menu: Simplify\342\211\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simplify( );
JSFHThe eigenfunctions are therefore multiples of 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.JSFH<Text-field style="Heading 2" layout="Heading 2">Homogeneous Neumann Conditions</Text-field>JSFHSubjecting , the solution of the differential equation , to the homogeneous Neumann conditions 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 results in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEkX0MyRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSI9RidGMi9GNlEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JVEiMEYnRjJGPEYvRjJGNQ== and LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEkX0MxRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= undetermined. Hence, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjBGJ0YyRjRGMkY0 is now an eigenvalue, with eigenfunction 1.For the general case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSsmTm90RXF1YWw7RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjBGJ0YyRjRGMkY0, the solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= remains a valid starting point.
Apply the Neumann conditionsApply the appropriate condition and press the Enter key.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 the equations in Matrix form, set determinant to zero, and solveReference equations via equation labels and press the Enter key.Context Menu: Student Linear Algebra\342\211\273Constructions\342\211\273Generate Matrix\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EkX0MxRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiLEYnRjJGNS9GOVEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYnUSRfQzJGJ0YvRjJGNUY4RjJGPw==Context Menu: Select Element\342\211\2731Context Menu: Student Linear Algebra\342\211\273Standard Operations\342\211\273DeterminantContext Menu: Solve\342\211\273Solve (general solution)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Student:-LinearAlgebra:-GenerateMatrix( [ ], [_C1,_C2 ], augmented=false );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRL3NlbGVjdH5lbnRyeX4xRidGL0YyRjVGRQ==()[1];LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRLGRldGVybWluYW50RidGL0YyRjVGRQ==Student:-LinearAlgebra:-Determinant( );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRJnNvbHZlRidGL0YyRjVGRQ==solve( { = 0 }, AllSolutions=true );
JSFHThe eigenvalue LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjBGJ0YyRjRGMkY0 is contained in the general solution, and the nonzero eigenvalues are again 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, where n is a positive integer. The corresponding eigenfunctions are found next with calculations similar to those employed for the Dirichlet conditions. However, for the Neumann conditions, collect the sine terms, then differentiate with respect to x.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Again apply the Neumann conditions to the solution in the form of , noting that is its derivative.
Apply the boundary conditions to Expression palette: Evaluation template
Press the Enter key.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Expression palette: Evaluation template
(Include result from first condition.)
Press the Enter key.Context Menu: Simplify\342\211\273Assuming 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( , 'assume=integer' );
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=From , deduce that the eigenfunctions are multiples of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2Jy1GLDYmUSJuRicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJy1JI21vR0YkNi5RIn5GJ0YyRjQvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZXLUYsNiZRInhGJ0Y/RjJGQUYyRjRGMkY0RjJGNA==.<Text-field style="Heading 1" layout="Heading 1">Example: An ODE That Is Not Self-Adjoint</Text-field>JSFHConsider the eigenvalue problem defined by the differential equation 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, which is not formally self-adjoint, and the homogeneous boundary conditionsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbW5HRiQ2JVEiM0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSSNtaUdGJDYmUSJ5RicvJSdpdGFsaWNHUSV0cnVlRidGLy9GM1EnaXRhbGljRictRjY2LlEoJnByaW1lO0YnRi9GMkY5RjtGPUY/RkFGQ0ZFL0ZIUSwwLjExMTExMTFlbUYnRkotSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYlUSIwRidGL0YyRi9GMkYvRjItRjY2LlEiK0YnRi9GMkY5RjtGPUY/RkFGQ0ZFL0ZIUSwwLjIyMjIyMjJlbUYnL0ZLRmBvLUYsNiVRIjVGJ0YvRjJGNUZMRlotRjY2LlEiPUYnRi9GMkY5RjtGPUY/RkFGQ0ZFL0ZIUSwwLjI3Nzc3NzhlbUYnL0ZLRmlvRmluRi9GMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbW5HRiQ2JVEiMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSSNtaUdGJDYmUSJ5RicvJSdpdGFsaWNHUSV0cnVlRidGLy9GM1EnaXRhbGljRictRjY2LlEoJnByaW1lO0YnRi9GMkY5RjtGPUY/RkFGQ0ZFL0ZIUSwwLjExMTExMTFlbUYnRkotSShtZmVuY2VkR0YkNiUtRiM2JS1GTTYmUScmIzk2MDtGJy9GUUYxRi9GMkYvRjJGL0YyLUY2Ni5RIitGJ0YvRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yMjIyMjIyZW1GJy9GS0Zhb0ZMRlotRjY2LlEiPUYnRi9GMkY5RjtGPUY/RkFGQ0ZFL0ZIUSwwLjI3Nzc3NzhlbUYnL0ZLRmdvLUYsNiVRIjBGJ0YvRjJGL0YyThe characteristic equation, obtained by making the exponential guess 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, is 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, with solutions 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. Thus, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjFGJ0YyRjRGMkY0 means 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 is a double root, and for that case, 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. To this solution apply the boundary conditions to see if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjFGJ0YyRjRGMkY0 is an eigenvalue.The work-flow here is to make the name S available so it can become the name of the solution, then to apply the boundary conditions to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=, and to examine the resulting homogeneous equations in matrix form. If the determinant of the resulting algebraic system is nonzero, then the only solution will be the null solution, and hence, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RIj1GJ0YyRkAvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWLUkjbW5HRiQ2JVEiMEYnRjJGQEYyRkA= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjFGJ0YyRjRGMkY0 is not an eigenvalue.
Context Menu: UnassignLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRKXVuYXNzaWduRidGL0YyRjVGRQ==unassign(''); ;Context Menu: Assign 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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRM2Fzc2lnbn5hc35mdW5jdGlvbkYnRi9GMkY1RkU=proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );Apply the boundary conditions to the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= and press the Enter key.Context Menu: Student Linear Algebra\342\211\273Constructions\342\211\273Generate Matrix\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEiYkYnRi9GMkY1RjJGPA==Context Menu: Select Element\342\211\2731Context Menu: Student Linear Algebra\342\211\273Standard Operations\342\211\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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRL3Rvfk1hdHJpeH5mb3JtRidGL0YyRjVGRQ==Student:-LinearAlgebra:-GenerateMatrix( [ ], [a,b ], augmented=false );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRL3NlbGVjdH5lbnRyeX4xRidGL0YyRjVGRQ==()[1];LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRLGRldGVybWluYW50RidGL0YyRjVGRQ==Student:-LinearAlgebra:-Determinant( );
Next, apply the boundary conditions to the solution of the ODE when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSsmTm90RXF1YWw7RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjFGJ0YyRjRGMkY0. The resulting homogeneous equations are cast into matrix form, and the determinant of the system matrix is set equal to zero. The eigenvalues are any values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 for which this determinant is zero.
Tools\342\211\273Load Package:
Student Calculus 1Loading Student:-Calculus1with(Student:-Calculus1):Context Menu: UnassignLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRKXVuYXNzaWduRidGL0YyRjVGRQ==unassign(''); ;Context Menu: Assign 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proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );Apply the boundary conditions to the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= and press the Enter key.Context Menu: Student Linear Algebra\342\211\273Constructions\342\211\273Generate Matrix\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEiYkYnRi9GMkY1RjJGPA==Context Menu: Select Element\342\211\2731Context Menu: Student Linear Algebra\342\211\273Standard Operations\342\211\273DeterminantContext Menu: Simplify\342\211\273SimplifyContext Menu: Collect\342\211\273Function\342\211\273(Select the first exponential expression as per Figure 2.)Context Menu: Student Calculus1\342\211\273Solve\342\211\273Find Roots\342\211\273(Complete the dialog as per Figure 3.)Context Menu: Assign to a Name\342\211\273Lambda
The six eigenvalues in the list LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5MjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 can only be obtained numerically. Note that the first eigenvalue is less than 1, whereas the remaining ones are greater than 1. This means that in the expression for the eigenfunctions, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkmbXNxcnRHRiQ2Iy1GIzYnLUkjbW5HRiQ2JVEiMUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEoJm1pbnVzO0YnRjpGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtRiw2JlEnJiM5NTU7RicvJSdpdGFsaWNHRjxGOkY9RjpGPUY6Rj1GK0Y6Rj0= will be real for the first eigenvalue and complex for all the others. This has implications for the corresponding eigenfunctions. To find these eigenfunctions, first solve the first boundary condition for a, and replace a in the general solution. Since b is then an arbitrary multiple, set it to 1. The resulting form of the solution will yield the first eigenfunction corresponding to the eigenvalue that is less than 1.
Apply the first boundary condition to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= and press the Enter key.Context Menu: Solve\342\211\273Isolate Expression for\342\211\273a Context Menu: Substitute Into\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=Context Menu: Evaluate at a Point\342\211\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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRLmlzb2xhdGV+Zm9yfmFGJ0YvRjJGNUZFisolate( , a );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMHN1YnN0aXR1dGV+aW50b0YnRi9GMkY1RkU=eval( S(x), );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMmV2YWx1YXRlfmF0fnBvaW50RidGL0YyRjVGRQ==eval( , [b = 1]);
For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI+RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiVRIjFGJ0YyRjRGMkY0, is complex. Both the real and imaginary parts are solutions of the ODE, but these two solutions can't be independent - they must be proportional. Hence, select, say, the real part via
JSFHNote the switch to explicit Maple commands. While SR could have been obtained with the Context Menu, the collect option used next provides for the simultaneous collection of terms, a feature not available in a syntax-free form.
Divide SR by its value at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 and apply the collect command to obtain a still simpler form.
JSFHConstruct Figure 4, a graph of these six eigenfunctions.
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 4 The first six eigenfunctions
JSFH<Text-field style="Heading 1" layout="Heading 1">A Numeric Approach</Text-field>JSFHSurprisingly, Maple's dsolve command is capable of computing eigenpairs numerically.JSFH
InitializeContext Menu: Assign to a Name\342\211\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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMWFzc2lnbn50b35hfm5hbWVGJ0YvRjJGNUZFDE := : 'DE';Context Menu: Assign to a Name\342\211\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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMWFzc2lnbn50b35hfm5hbWVGJ0YvRjJGNUZFBC[1] := : 'BC[1]';Context Menu: Assign to a Name\342\211\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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMWFzc2lnbn50b35hfm5hbWVGJ0YvRjJGNUZFBC[2] := : 'BC[2]';Context Menu: Assign to a Name\342\211\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LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMWFzc2lnbn50b35hfm5hbWVGJ0YvRjJGNUZFsys := : 'sys';
JSFHTable 2 shows how to obtain eigenpairs numerically. The left-hand column shows how to apply the dsolve command; the right, how to extract the computed eigenvalue. In order for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 to be an unknown in these calculations, an additional constraint must be provided. This is the normalization LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSNtbkdGJDYlUSIwRidGMi9GNlEnbm9ybWFsRidGMkZBRjJGQS1JI21vR0YkNi5RIj1GJ0YyRkEvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUY+NiVRIjFGJ0YyRkFGMkZB. In addition, good approximations of both the shape of the eigenfunction and the eigenvalue must be provided. This latter is the chief obstacle to this method becoming a practical tool. These approximations become harder and harder to determine. In effect, one needs to have the solution in order to calculate the solution.JSFHJSFH
JSFHJSFHLegal Notice: \302\251 Maplesoft, a division of Waterloo Maple Inc. 2017. Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities.JSFHJSFHJSFHJSFHJSFHJSFHJSFHJSFHJSFHJSFHJSFHJSFH