85.12.9 problem 9

Internal problem ID [22503]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 40
Problem number : 9
Date solved : Thursday, October 02, 2025 at 08:43:29 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 55

ode:=x^3+y(x)^3-x*y(x)^2*diff(y(x),x) = 0; 
ic:=[y(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= 3^{{1}/{3}} \ln \left (x \right )^{{1}/{3}} x \\ y &= \frac {\ln \left (x \right )^{{1}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x}{2} \\ y &= -\frac {\ln \left (x \right )^{{1}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 58

ode=(x^3+y[x]^3)-(x*y[x]^2)*D[y[x],x]==0; 
ic={y[1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.972 (sec). Leaf size: 73

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 = {y(1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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