61.30.1 problem 149

Internal problem ID [12649]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 149
Date solved : Tuesday, January 28, 2025 at 08:03:11 PM
CAS classification : [_Gegenbauer]

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

✓ Solution by Maple

Time used: 0.278 (sec). Leaf size: 52

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

\[ y = -\left (x -1\right ) \left (x +1\right ) \left (\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, \frac {3}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) c_{2} x +c_{1} \operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.108 (sec). Leaf size: 56

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

\[ y(x)\to i c_2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},\frac {n}{2},\frac {3}{2},x^2\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {n-1}{2},-\frac {n}{2},\frac {1}{2},x^2\right ) \]