Internal problem ID [2021]
Book: Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition,
2002
Section: Chapter 16, Series solutions of ODEs. Section 16.6 Exercises, page 550
Problem number: Problem 16.1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
Solve \begin {gather*} \boxed {\left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+\lambda y=0} \end {gather*} With the expansion point for the power series method at \(z = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 63
Order:=6; dsolve((1-z^2)*diff(y(z),z$2)-3*z*diff(y(z),z)+lambda*y(z)=0,y(z),type='series',z=0);
\[ y \relax (z ) = \left (1-\frac {\lambda \,z^{2}}{2}+\frac {\lambda \left (\lambda -8\right ) z^{4}}{24}\right ) y \relax (0)+\left (z -\frac {\left (\lambda -3\right ) z^{3}}{6}+\frac {\left (\lambda -3\right ) \left (\lambda -15\right ) z^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (z^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 80
AsymptoticDSolveValue[(1-z^2)*y''[z]-3*z*y'[z]+\[Lambda]*y[z]==0,y[z],{z,0,5}]
\[ y(z)\to c_2 \left (\frac {\lambda ^2 z^5}{120}-\frac {3 \lambda z^5}{20}+\frac {3 z^5}{8}-\frac {\lambda z^3}{6}+\frac {z^3}{2}+z\right )+c_1 \left (\frac {\lambda ^2 z^4}{24}-\frac {\lambda z^4}{3}-\frac {\lambda z^2}{2}+1\right ) \]