59.1.634 problem 651

Internal problem ID [9806]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 651
Date solved : Monday, January 27, 2025 at 06:14:29 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.803 (sec). Leaf size: 45

dsolve((1-z^2)*diff(y(z),z$2)-3*z*diff(y(z),z)+y(z)=0,y(z), singsol=all)
 

\[ y \left (z \right ) = \frac {c_{1} \left (z +\sqrt {z^{2}-1}\right )^{\sqrt {2}}+c_{2} \left (z +\sqrt {z^{2}-1}\right )^{-\sqrt {2}}}{\sqrt {z^{2}-1}} \]

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 90

DSolve[(1-z^2)*D[y[z],{z,2}]-3*z*D[y[z],z]+y[z]==0,y[z],z,IncludeSingularSolutions -> True]
 

\[ y(z)\to \frac {\sqrt {2} c_1 \cos \left (2 \sqrt {2} \arcsin \left (\frac {\sqrt {1-z}}{\sqrt {2}}\right )\right )+\sqrt {\pi } c_2 \sqrt [4]{1-z^2} Q_{-\frac {1}{2}+\sqrt {2}}^{\frac {1}{2}}(z)}{\sqrt {\pi } \sqrt [4]{-\left (z^2-1\right )^2}} \]