55.32.14 problem 195

Internal problem ID [13968]
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 : 195
Date solved : Friday, October 03, 2025 at 07:04:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 1.538 (sec). Leaf size: 1493

ode:=(a*x^3+b*x^2+c*x)*diff(diff(y(x),x),x)+(alpha*x^2+beta*x+2*c)*diff(y(x),x)+(beta-2*b)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 2.667 (sec). Leaf size: 139

ode=(a*x^3+b*x^2+c*x)*D[y[x],{x,2}]+(\[Alpha]*x^2+\[Beta]*x+2*c)*D[y[x],x]+(\[Beta]-2*b)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {(2 a x+2 b-\beta -\alpha x) \left (c_2 \int _1^x\frac {\exp \left (\frac {(b \alpha +2 a (b-\beta )) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right ) (c+K[1] (b+a K[1]))^{1-\frac {\alpha }{2 a}}}{(-2 b+\beta +(\alpha -2 a) K[1])^2}dK[1]+c_1\right )}{x (2 b-\beta )} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq((BETA - 2*b)*y(x) + (Alpha*x**2 + BETA*x + 2*c)*Derivative(y(x), x) + (a*x**3 + b*x**2 + c*x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False