61.31.8 problem 189

Internal problem ID [12689]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 189
Date solved : Tuesday, January 28, 2025 at 03:39:04 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 46

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

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

✓ Solution by Mathematica

Time used: 0.189 (sec). Leaf size: 39

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

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