Internal problem ID [7370]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 651.
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 }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 51
dsolve((1-z^2)*diff(y(z),z$2)-3*z*diff(y(z),z)+y(z)=0,y(z), singsol=all)
\[ y \relax (z ) = \frac {c_{1} \left (z +\sqrt {z^{2}-1}\right )^{\sqrt {2}}}{\sqrt {z^{2}-1}}+\frac {c_{2} \left (z +\sqrt {z^{2}-1}\right )^{-\sqrt {2}}}{\sqrt {z^{2}-1}} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 69
DSolve[(1-z^2)*y''[z]-3*z*y'[z]+y[z]==0,y[z],z,IncludeSingularSolutions -> True]
\begin{align*} y(z)\to \frac {\sqrt {\frac {2}{\pi }} c_1 \cos \left (\sqrt {2} \text {ArcCos}(z)\right )+c_2 \sqrt [4]{1-z^2} Q_{-\frac {1}{2}+\sqrt {2}}^{\frac {1}{2}}(z)}{\sqrt [4]{-\left (z^2-1\right )^2}} \\ \end{align*}