61.32.17 problem 226

Internal problem ID [12727]
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 : 226
Date solved : Tuesday, January 28, 2025 at 08:23:52 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 )-\mu ^{2}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.198 (sec). Leaf size: 17

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

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

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 20

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

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