55.25.36 problem 36

Internal problem ID [13745]
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 : 36
Date solved : Sunday, October 12, 2025 at 05:24:56 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 2162

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

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

✓ Mathematica. Time used: 6.05 (sec). Leaf size: 67

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

\[ \text {Solve}\left [-\frac {c \left (\log \left (a^2 (n-1) x^n+a x (b (n-1) x+(n-2) y(x))+c x^2\right )+(n-2) \log (c x-a y(x))-n \log (x)\right )}{n-2}=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out