Internal problem ID [2034]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {\left (-z^{2}+1\right ) y^{\prime \prime }-z y^{\prime }+m^{2} y=0} \end {gather*} With the expansion point for the power series method at \(z = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 71
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);
\[ y \relax (z ) = \left (1-\frac {m^{2} z^{2}}{2}+\frac {m^{2} \left (m^{2}-4\right ) z^{4}}{24}\right ) y \relax (0)+\left (z +\frac {\left (-m^{2}+1\right ) z^{3}}{6}+\frac {\left (m^{4}-10 m^{2}+9\right ) z^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (z^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 88
AsymptoticDSolveValue[(1-z^2)*y''[z]-z*y'[z]+m^2*y[z]==0,y[z],{z,0,5}]
\[ y(z)\to c_2 \left (\frac {m^4 z^5}{120}-\frac {m^2 z^5}{12}-\frac {m^2 z^3}{6}+\frac {3 z^5}{40}+\frac {z^3}{6}+z\right )+c_1 \left (\frac {m^4 z^4}{24}-\frac {m^2 z^4}{6}-\frac {m^2 z^2}{2}+1\right ) \]