50.5.10 problem 1(j)

Internal problem ID [7880]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.7. Homogeneous Equations. Page 28
Problem number : 1(j)
Date solved : Sunday, March 30, 2025 at 12:35:20 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 56

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

\begin{align*} y &= \left (3 \ln \left (x \right )+c_1 \right )^{{1}/{3}} x \\ y &= -\frac {\left (3 \ln \left (x \right )+c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2} \\ y &= \frac {\left (3 \ln \left (x \right )+c_1 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.192 (sec). Leaf size: 63

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

\begin{align*} y(x)\to x \sqrt [3]{3 \log (x)+c_1} \\ y(x)\to -\sqrt [3]{-1} x \sqrt [3]{3 \log (x)+c_1} \\ y(x)\to (-1)^{2/3} x \sqrt [3]{3 \log (x)+c_1} \\ \end{align*}

✓ Sympy. Time used: 1.498 (sec). Leaf size: 68

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

\[ \left [ y{\left (x \right )} = \sqrt [3]{x^{3} \left (C_{1} + 3 \log {\left (x \right )}\right )}, \ y{\left (x \right )} = \frac {\sqrt [3]{x^{3} \left (C_{1} + 3 \log {\left (x \right )}\right )} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{x^{3} \left (C_{1} + 3 \log {\left (x \right )}\right )} \left (-1 + \sqrt {3} i\right )}{2}\right ] \]