Internal problem ID [8126]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 651.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
\[ \boxed {\left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (z \right ) = \frac {c_{1} \left (z +\sqrt {z^{2}-1}\right )^{\sqrt {2}}}{\sqrt {z^{2}-1}}+\frac {c_{2} \left (z +\sqrt {z^{2}-1}\right )^{-\sqrt {2}}}{\sqrt {z^{2}-1}} \]
✓ Solution by Mathematica
Time used: 0.071 (sec). Leaf size: 90
DSolve[(1-z^2)*y''[z]-3*z*y'[z]+y[z]==0,y[z],z,IncludeSingularSolutions -> True]
\[ y(z)\to \frac {\sqrt {2} c_1 \cos \left (2 \sqrt {2} \arcsin \left (\frac {\sqrt {1-z}}{\sqrt {2}}\right )\right )+\sqrt {\pi } c_2 \sqrt [4]{1-z^2} Q_{-\frac {1}{2}+\sqrt {2}}^{\frac {1}{2}}(z)}{\sqrt {\pi } \sqrt [4]{-\left (z^2-1\right )^2}} \]