85.19.2 problem 2

Internal problem ID [22559]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. B Exercises at page 52
Problem number : 2
Date solved : Thursday, October 02, 2025 at 08:51:27 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y^{\prime }&=\frac {x}{x^{2} y+y^{3}} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 55

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

\begin{align*} y &= \sqrt {-1-\operatorname {LambertW}\left (-2 c_1 \,{\mathrm e}^{-x^{2}-1}\right )-x^{2}} \\ y &= -\sqrt {-1-\operatorname {LambertW}\left (-2 c_1 \,{\mathrm e}^{-x^{2}-1}\right )-x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 60.069 (sec). Leaf size: 65

ode=D[y[x],x]==x/( x^2*y[x]+y[x]^3  ); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 2.016 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x/(x**2*y(x) + y(x)**3) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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