61.31.11 problem 192

Internal problem ID [12692]
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 : 192
Date solved : Tuesday, January 28, 2025 at 03:39:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (a x +b \right ) y^{\prime \prime }-2 x \left (a x +2 b \right ) y^{\prime }+2 \left (a x +3 b \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 20

dsolve(x^2*(a*x+b)*diff(y(x),x$2)-2*x*(a*x+2*b)*diff(y(x),x)+2*(a*x+3*b)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.199 (sec). Leaf size: 44

DSolve[x^2*(a*x+b)*D[y[x],{x,2}]-2*x*(a*x+2*b)*D[y[x],x]+2*(a*x+3*b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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