61.30.4 problem 152

Internal problem ID [12652]
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 : 152
Date solved : Tuesday, January 28, 2025 at 03:24:25 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 45

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

\[ y(x)\to c_1 \cosh \left (n \log \left (\sqrt {x^2-1}+x\right )\right )+i c_2 \sinh \left (n \log \left (\sqrt {x^2-1}+x\right )\right ) \]