61.30.8 problem 156

Internal problem ID [12656]
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 : 156
Date solved : Tuesday, January 28, 2025 at 08:03:16 PM
CAS classification : [_Gegenbauer]

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

✓ Solution by Maple

Time used: 0.204 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 32

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

\[ y(x)\to \left (x^2-1\right )^{-n/2} (c_1 P_{\nu }^n(x)+c_2 Q_{\nu }^n(x)) \]