28.1.103 problem 126

Internal problem ID [4409]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 126
Date solved : Sunday, March 30, 2025 at 03:17:55 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y^{\prime }&=\frac {1}{x y+x^{3} y^{3}} \end{align*}

✓ Maple. Time used: 0.102 (sec). Leaf size: 68

ode:=diff(y(x),x) = 1/(x*y(x)+x^3*y(x)^3); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 60.135 (sec). Leaf size: 68

ode=D[y[x],x]==1/(x*y[x]+x^3*y[x]^3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {x^2 W\left (8 c_1 e^{\frac {1}{x^2}-1}\right )+x^2-1}}{x} \\ y(x)\to \frac {\sqrt {x^2 W\left (8 c_1 e^{\frac {1}{x^2}-1}\right )+x^2-1}}{x} \\ \end{align*}

✓ Sympy. Time used: 0.883 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - 1/(x**3*y(x)**3 + x*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \frac {\left (y^{2}{\left (x \right )} - 1\right ) e^{y^{2}{\left (x \right )}}}{2} - \frac {e^{y^{2}{\left (x \right )}}}{2 x^{2}} = 0 \]