81.1.24 problem 2-22

Internal problem ID [21469]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 2. Separable differential equations
Problem number : 2-22
Date solved : Thursday, October 02, 2025 at 07:39:08 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 50

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

\begin{align*} y &= \frac {\sqrt {\left (x^{2}+1\right ) \left (x^{2}+c_1 \right )}}{x^{2}+1} \\ y &= -\frac {\sqrt {\left (x^{2}+1\right ) \left (x^{2}+c_1 \right )}}{x^{2}+1} \\ \end{align*}

✓ Mathematica. Time used: 0.242 (sec). Leaf size: 70

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

\begin{align*} y(x)&\to -\frac {\sqrt {x^2+1+e^{2 c_1}}}{\sqrt {x^2+1}}\\ y(x)&\to \frac {\sqrt {x^2+1+e^{2 c_1}}}{\sqrt {x^2+1}}\\ y(x)&\to -1\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.433 (sec). Leaf size: 32

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

\[ \left [ y{\left (x \right )} = - \sqrt {\frac {C_{1} + x^{2}}{x^{2} + 1}}, \ y{\left (x \right )} = \sqrt {\frac {C_{1} + x^{2}}{x^{2} + 1}}\right ] \]