55.32.15 problem 196

Internal problem ID [13969]
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 : 196
Date solved : Friday, October 03, 2025 at 07:04:11 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 (\alpha x +2 b -\beta \right ) y&=0 \end{align*}

✓ Maple. Time used: 1.371 (sec). Leaf size: 1497

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)-(alpha*x+2*b-beta)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 3.549 (sec). Leaf size: 224

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

\begin{align*} y(x)&\to \frac {\left (b (2 a x-3 \beta -2 \alpha x)-a \alpha x^2-2 a \beta x+2 b^2+\beta ^2+\alpha c+\alpha ^2 x^2+2 \alpha \beta x\right ) \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}}}{\left (2 b^2-3 \beta b+2 (a-\alpha ) K[1] b+\beta ^2+\alpha ^2 K[1]^2-a \alpha K[1]^2+c \alpha -2 a \beta K[1]+2 \alpha \beta K[1]\right )^2}dK[1]+c_1\right )}{x \left (2 b^2+\beta ^2-3 \beta b+\alpha c\right )} \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((-Alpha*x + 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