61.30.17 problem 165

Internal problem ID [12665]
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 : 165
Date solved : Tuesday, January 28, 2025 at 03:31:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.314 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.641 (sec). Leaf size: 109

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

\[ y(x)\to \left (a^2+x^2\right )^{-b/2} \exp \left (\int _1^x\frac {i \sqrt {(b-1)^2} a+K[1]}{a^2+K[1]^2}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {i \sqrt {(b-1)^2} a+K[1]}{a^2+K[1]^2}dK[1]\right )dK[2]+c_1\right ) \]