61.32.6 problem 216

Internal problem ID [12716]
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 : 216
Date solved : Tuesday, January 28, 2025 at 04:16:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 67

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

\[ y = \sqrt {x \left (a -x \right )}\, \left (\left (\frac {x}{a -x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{2} +\left (\frac {a -x}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.224 (sec). Leaf size: 112

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

\[ y(x)\to \exp \left (\int _1^x\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[2]+c_1\right ) \]