55.26.19 problem 19

Internal problem ID [13766]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.4. Equations Containing Polynomial Functions of y. subsection 1.4.1-2 Abel equations of the first kind.
Problem number : 19
Date solved : Thursday, October 02, 2025 at 08:00:32 AM
CAS classification : [_Abel]

\begin{align*} y^{\prime }&=\frac {y^{3}}{\sqrt {a \,x^{2}+b x +c}}+y^{2} \end{align*}

✗ Maple

ode:=diff(y(x),x) = 1/(a*x^2+b*x+c)^(1/2)*y(x)^3+y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==(a*x^2+b*x+c)^(-1/2)*y[x]^3+y[x]^2; 
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") 
y = Function("y") 
ode = Eq(-y(x)**2 + Derivative(y(x), x) - y(x)**3/sqrt(a*x**2 + b*x + c),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out