70.5.3 problem 3

Internal problem ID [18710]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 3
Date solved : Thursday, October 02, 2025 at 03:22:17 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \frac {\left (3 x^{3}-x y^{2}\right ) y^{\prime }}{y^{3}+3 x^{2} y}&=1 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 22

ode:=(3*x^3-x*y(x)^2)/(3*x^2*y(x)+y(x)^3)*diff(y(x),x) = 1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {3}\, \sqrt {-\frac {1}{\operatorname {LambertW}\left (-3 c_1 \,x^{4}\right )}}\, x \]

✓ Mathematica. Time used: 0.064 (sec). Leaf size: 61

ode=(3*x^3-x*y[x]^2)/(3*x^2*y[x]+y[x]^3)*D[y[x],x]==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i \sqrt {3} x}{\sqrt {W\left (-3 e^{-2 c_1} x^4\right )}}\\ y(x)&\to \frac {i \sqrt {3} x}{\sqrt {W\left (-3 e^{-2 c_1} x^4\right )}} \end{align*}

✓ Sympy. Time used: 1.462 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x**3 - x*y(x)**2)*Derivative(y(x), x)/(3*x**2*y(x) + y(x)**3) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {e^{2 C_{1} + \frac {W\left (- 3 x^{4} e^{- 4 C_{1}}\right )}{2}}}{x} \]