26.4.1 problem Recognizable Exact Differential equations. Integrating factors. Example 10.51, page 90

Internal problem ID [6947]
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. Example 10.51, page 90
Date solved : Tuesday, September 30, 2025 at 04:07:09 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 13

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

\[ y = \frac {x}{-x +c_1} \]

✓ Mathematica. Time used: 0.133 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] (K[1]+1)}dK[1]\&\right ][\log (x)+c_1]\\ y(x)&\to -1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.158 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = - \frac {x e^{C_{1}}}{x e^{C_{1}} - 1} \]