55.25.25 problem 25

Internal problem ID [13734]
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 : 25
Date solved : Thursday, October 02, 2025 at 05:44:14 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✗ Maple

ode:=(2*A*x*y(x)-A*k*x^2+a*y(x)+b*x+c)*diff(y(x),x) = A*y(x)^2+m*y(x)+k*(a*k+b-m)*x+s; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

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

Timed out

✗ Sympy

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

Timed Out