18.3.14 problem Problem 16.15

Internal problem ID [3514]
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 16.15
Date solved : Monday, January 27, 2025 at 07:40:09 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 65

Order:=6; 
dsolve((1-z^2)*diff(y(z),z$2)-z*diff(y(z),z)+m^2*y(z)=0,y(z),type='series',z=0);
 

\[ y \left (z \right ) = \left (1-\frac {m^{2} z^{2}}{2}+\frac {m^{2} \left (m^{2}-4\right ) z^{4}}{24}\right ) y \left (0\right )+\left (z -\frac {\left (m^{2}-1\right ) z^{3}}{6}+\frac {\left (m^{4}-10 m^{2}+9\right ) z^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (z^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 88

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

\[ y(z)\to c_2 \left (\frac {m^4 z^5}{120}-\frac {m^2 z^5}{12}-\frac {m^2 z^3}{6}+\frac {3 z^5}{40}+\frac {z^3}{6}+z\right )+c_1 \left (\frac {m^4 z^4}{24}-\frac {m^2 z^4}{6}-\frac {m^2 z^2}{2}+1\right ) \]