65.7.12 problem 16

Internal problem ID [15720]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.3.3, page 71
Problem number : 16
Date solved : Thursday, October 02, 2025 at 10:24:02 AM
CAS classification : [_separable]

\begin{align*} x \left (1-y^{3}\right )-3 y^{2} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 64

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

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

✓ Mathematica. Time used: 1.812 (sec). Leaf size: 111

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

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

✓ Sympy. Time used: 1.417 (sec). Leaf size: 68

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

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