55.25.34 problem 34

Internal problem ID [13743]
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 : 34
Date solved : Sunday, October 12, 2025 at 05:17:38 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} x \left (\left (m -1\right ) \left (A x +B \right ) y+m \left (d \,x^{2}+e x +F \right )\right ) y^{\prime }&=\left (A \left (1-n \right ) x -B n \right ) y^{2}+\left (d \left (2-n \right ) x^{2}+e \left (1-n \right ) x -F n \right ) y \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 2371

ode:=x*((m-1)*(A*x+B)*y(x)+m*(d*x^2+e*x+F))*diff(y(x),x) = (A*(-n+1)*x-B*n)*y(x)^2+(d*(-n+2)*x^2+e*(-n+1)*x-F*n)*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 143.711 (sec). Leaf size: 41

ode=x*( (m-1)*(A*x+B)*y[x]+m*(d*x^2+e*x+F) )*D[y[x],x]==( A*(1-n)*x-B*n)*y[x]^2+ (d*(2-n)*x^2+e*(1-n)*x-F*n)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [-A \log \left (A x y(x)+B y(x)+d x^2+e x+F\right )+A m \log (y(x))+A n \log (x)=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
m = symbols("m") 
A = symbols("A") 
B = symbols("B") 
d = symbols("d") 
e = symbols("e") 
F = symbols("F") 
n = symbols("n") 
y = Function("y") 
ode = Eq(x*(m*(F + d*x**2 + e*x) + (m - 1)*(A*x + B)*y(x))*Derivative(y(x), x) - (A*x*(1 - n) - B*n)*y(x)**2 - (-F*n + d*x**2*(2 - n) + e*x*(1 - n))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out