55.25.27 problem 27

Internal problem ID [13736]
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 : 27
Date solved : Thursday, October 02, 2025 at 06:24:03 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=(2*A*x*y[x]+B*x^2+a*y[x]+b*x+c)*D[y[x],x]==A*y[x]^2+k*(A*k+B)*x^2+b*y[x]+a*k^2*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") 
s = symbols("s") 
y = Function("y") 
ode = Eq(-A*y(x)**2 - a*k**2*x - b*y(x) - k*x**2*(A*k + B) - s + (2*A*x*y(x) + B*x**2 + a*y(x) + b*x + c)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out