61.30.6 problem 154

Internal problem ID [12654]
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 : 154
Date solved : Tuesday, January 28, 2025 at 08:03:14 PM
CAS classification : [_Gegenbauer]

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

✓ Solution by Maple

Time used: 0.189 (sec). Leaf size: 15

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

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

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 18

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

\[ y(x)\to c_1 \operatorname {LegendreP}(\nu ,x)+c_2 \operatorname {LegendreQ}(\nu ,x) \]