81.3.3 problem 4-8

Internal problem ID [21502]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 4. Homogeneous differential equations.
Problem number : 4-8
Date solved : Thursday, October 02, 2025 at 07:42:26 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 58

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

\begin{align*} y &= \left (-6 \ln \left (x \right )+c_1 \right )^{{1}/{3}} x \\ y &= -\frac {\left (-6 \ln \left (x \right )+c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2} \\ y &= \frac {\left (-6 \ln \left (x \right )+c_1 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 63

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

\begin{align*} y(x)&\to x \sqrt [3]{-6 \log (x)+c_1}\\ y(x)&\to -\sqrt [3]{-1} x \sqrt [3]{-6 \log (x)+c_1}\\ y(x)&\to (-1)^{2/3} x \sqrt [3]{-6 \log (x)+c_1} \end{align*}

✓ Sympy. Time used: 0.949 (sec). Leaf size: 68

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

\[ \left [ y{\left (x \right )} = \sqrt [3]{x^{3} \left (C_{1} - 6 \log {\left (x \right )}\right )}, \ y{\left (x \right )} = \frac {\sqrt [3]{x^{3} \left (C_{1} - 6 \log {\left (x \right )}\right )} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{x^{3} \left (C_{1} - 6 \log {\left (x \right )}\right )} \left (-1 + \sqrt {3} i\right )}{2}\right ] \]