55.33.20 problem 229

Internal problem ID [14003]
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 : 229
Date solved : Thursday, October 02, 2025 at 09:08:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.206 (sec). Leaf size: 817

ode:=(a*x^2+b)^2*diff(diff(y(x),x),x)+(a*x^2+b)*(c*x^2+d)*diff(y(x),x)+2*(-a*d+b*c)*x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 60.09 (sec). Leaf size: 104

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

\begin{align*} y(x)&\to \exp \left (\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right ) (b c-a d)}{a^{3/2} \sqrt {b}}-\frac {c x}{a}\right ) \left (\int _1^x\exp \left (\frac {(a d-b c) \arctan \left (\frac {\sqrt {a} K[1]}{\sqrt {b}}\right )}{a^{3/2} \sqrt {b}}+\frac {c K[1]}{a}\right ) c_1dK[1]+c_2\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
y = Function("y") 
ode = Eq(x*(-2*a*d + 2*b*c)*y(x) + (a*x**2 + b)**2*Derivative(y(x), (x, 2)) + (a*x**2 + b)*(c*x**2 + d)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False