61.30.7 problem 155

Internal problem ID [12655]
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 : 155
Date solved : Tuesday, January 28, 2025 at 03:24:29 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-3 x y^{\prime }+n \left (n +2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.161 (sec). Leaf size: 68

dsolve((1-x^2)*diff(y(x),x$2)-3*x*diff(y(x),x)+n*(n+2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 42

DSolve[(1-x^2)*D[y[x],{x,2}]-3*x*D[y[x],x]+n*(n+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {c_1 P_{n+\frac {1}{2}}^{\frac {1}{2}}(x)+c_2 Q_{n+\frac {1}{2}}^{\frac {1}{2}}(x)}{\sqrt [4]{x^2-1}} \]