26.4.11 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.3, page 90

Internal problem ID [6957]
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 10.3, page 90
Date solved : Tuesday, September 30, 2025 at 04:07:16 PM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 49

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

\begin{align*} y &= -\frac {\sqrt {-18 x^{4}-24 x^{3}+36 c_1}}{6 x} \\ y &= \frac {\sqrt {-18 x^{4}-24 x^{3}+36 c_1}}{6 x} \\ \end{align*}

✓ Mathematica. Time used: 0.187 (sec). Leaf size: 139

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

\begin{align*} y(x)&\to -\frac {\sqrt {2 \int _1^x-K[1]^2 (K[1]+1)dK[1]+c_1}}{x}\\ y(x)&\to \frac {\sqrt {2 \int _1^x-K[1]^2 (K[1]+1)dK[1]+c_1}}{x}\\ y(x)&\to -\frac {\sqrt {2} \sqrt {\int _1^x-K[1]^2 (K[1]+1)dK[1]}}{x}\\ y(x)&\to \frac {\sqrt {2} \sqrt {\int _1^x-K[1]^2 (K[1]+1)dK[1]}}{x} \end{align*}

✓ Sympy. Time used: 0.303 (sec). Leaf size: 42

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {C_{1} - 18 x^{4} - 24 x^{3}}}{6 x}, \ y{\left (x \right )} = \frac {\sqrt {C_{1} - 18 x^{4} - 24 x^{3}}}{6 x}\right ] \]