Internal problem ID [6948]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 217.
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*}
✓ Solution by Maple
Time used: 0.328 (sec). Leaf size: 55
dsolve((1-z^2)*diff(y(z),z$2)-3*z*diff(y(z),z)+lambda*y(z)=0,y(z), singsol=all)
\[ y \relax (z ) = \frac {c_{1} \left (z +\sqrt {z^{2}-1}\right )^{\sqrt {\lambda +1}}}{\sqrt {z^{2}-1}}+\frac {c_{2} \left (z +\sqrt {z^{2}-1}\right )^{-\sqrt {\lambda +1}}}{\sqrt {z^{2}-1}} \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 54
DSolve[(1-z^2)*y''[z]-3*z*y'[z]+\[Lambda]*y[z]==0,y[z],z,IncludeSingularSolutions -> True]
\begin{align*} y(z)\to \frac {c_1 P_{\sqrt {\lambda +1}-\frac {1}{2}}^{\frac {1}{2}}(z)+c_2 Q_{\sqrt {\lambda +1}-\frac {1}{2}}^{\frac {1}{2}}(z)}{\sqrt [4]{z^2-1}} \\ \end{align*}