61.30.3 problem 151

Internal problem ID [12651]
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 : 151
Date solved : Tuesday, January 28, 2025 at 03:24:21 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 }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 50

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

\[ y(x)\to c_1 \cos \left (\sqrt {a} \log \left (\sqrt {x^2-1}+x\right )\right )+c_2 \sin \left (\sqrt {a} \log \left (\sqrt {x^2-1}+x\right )\right ) \]