26.4.25 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.17, page 90

Internal problem ID [6971]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number : Recognizable Exact Differential equations. Integrating factors. Exercise 10.17, page 90
Date solved : Tuesday, September 30, 2025 at 04:07:25 PM
CAS classification : [_rational]

\begin{align*} y-\left (y^{2}+x^{2}+x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 36

ode:=y(x)-(y(x)^2+x^2+x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {{\mathrm e}^{-2 i y} \left (i x +y\right )+2 c_1 \left (i y+x \right )}{2 i y+2 x} = 0 \]

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 91

ode=(y[x])-(y[x]^2+x^2+x)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {x}{x^2+K[2]^2}-\int _1^x\left (\frac {2 K[2]^2}{\left (K[1]^2+K[2]^2\right )^2}-\frac {1}{K[1]^2+K[2]^2}\right )dK[1]+1\right )dK[2]+\int _1^x-\frac {y(x)}{K[1]^2+y(x)^2}dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - y(x)/(x**2 + x + y(x)**2) cannot be solved by the factorable group method