61.25.6 problem 6

Internal problem ID [12420]
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 : 6
Date solved : Monday, March 31, 2025 at 05:33:53 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (c*x**n + s + (a*y(x) + b)*y(x))/(x*y(x)) cannot be solved by the factorable group method