59.1.214 problem 217

Internal problem ID [9386]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 217
Date solved : Monday, January 27, 2025 at 06:02:11 PM
CAS classification : [_Gegenbauer]

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

✓ Solution by Maple

Time used: 0.073 (sec). Leaf size: 49

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

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 54

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

\[ y(z)\to \frac {c_1 P_{\sqrt {\lambda +1}-\frac {1}{2}}^{\frac {1}{2}}(z)+c_2 Q_{\sqrt {\lambda +1}-\frac {1}{2}}^{\frac {1}{2}}(z)}{\sqrt [4]{z^2-1}} \]