61.30.9 problem 157

Internal problem ID [12657]
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 : 157
Date solved : Tuesday, January 28, 2025 at 08:03:17 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: 0.188 (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: 0.044 (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]
 

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