61.32.16 problem 225

Internal problem ID [12726]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 225
Date solved : Tuesday, January 28, 2025 at 08:23:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.203 (sec). Leaf size: 17

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

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

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 20

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

\[ y(x)\to c_1 P_{\nu }^n(x)+c_2 Q_{\nu }^n(x) \]