55.25.19 problem 19

Internal problem ID [13728]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.4-2.
Problem number : 19
Date solved : Thursday, October 02, 2025 at 04:36:40 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (2 A x y+B \,x^{2}+k x \right ) y^{\prime }&=A y^{2}+c x y+d \,x^{2}-c \beta x -A \,\beta ^{2}-k \beta \end{align*}

✗ Maple

ode:=(2*A*x*y(x)+B*x^2+k*x)*diff(y(x),x) = A*y(x)^2+c*x*y(x)+d*x^2-c*beta*x-A*beta^2-k*beta; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=(2*A*x*y[x]+B*x^2+k*x)*D[y[x],x]==A*y[x]^2+c*x*y[x]+d*x^2-c*\[Beta]*x-A*\[Beta]^2-k*\[Beta]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

Timed Out