87.6.5 problem 5

Internal problem ID [23322]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 53
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:32:04 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} -x^{2} y+\left (x^{3}+y^{3}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 16

ode:=-x^2*y(x)+(x^3+y(x)^3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\left (\frac {1}{\operatorname {LambertW}\left (c_1 \,x^{3}\right )}\right )}^{{1}/{3}} x \]

✓ Mathematica. Time used: 5.399 (sec). Leaf size: 80

ode=(-x^2*y[x])+(x^3+y[x]^3)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x}{\sqrt [3]{W\left (e^{-3 c_1} x^3\right )}}\\ y(x)&\to -\frac {\sqrt [3]{-1} x}{\sqrt [3]{W\left (e^{-3 c_1} x^3\right )}}\\ y(x)&\to \frac {(-1)^{2/3} x}{\sqrt [3]{W\left (e^{-3 c_1} x^3\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.490 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = e^{C_{1} + \frac {W\left (x^{3} e^{- 3 C_{1}}\right )}{3}} \]