18.3.1 problem Problem 16.1

Internal problem ID [3501]
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
Date solved : Monday, January 27, 2025 at 07:39:55 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+\lambda y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 53

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);
 

\[ y \left (z \right ) = \left (1-\frac {\lambda \,z^{2}}{2}+\frac {\lambda \left (\lambda -8\right ) z^{4}}{24}\right ) y \left (0\right )+\left (z -\frac {\left (\lambda -3\right ) z^{3}}{6}+\frac {\left (\lambda -3\right ) \left (\lambda -15\right ) z^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (z^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 80

AsymptoticDSolveValue[(1-z^2)*D[y[z],{z,2}]-3*z*D[y[z],z]+\[Lambda]*y[z]==0,y[z],{z,0,"6"-1}]
 

\[ y(z)\to c_2 \left (\frac {\lambda ^2 z^5}{120}-\frac {3 \lambda z^5}{20}+\frac {3 z^5}{8}-\frac {\lambda z^3}{6}+\frac {z^3}{2}+z\right )+c_1 \left (\frac {\lambda ^2 z^4}{24}-\frac {\lambda z^4}{3}-\frac {\lambda z^2}{2}+1\right ) \]