63.5.32 problem 16-b(i)

Internal problem ID [13035]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises page 41
Problem number : 16-b(i)
Date solved : Monday, March 31, 2025 at 07:31:52 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 66

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

\begin{align*} x &= 0 \\ x &= \frac {\left (-c_1 \,t^{2}\right )^{{1}/{3}}}{t} \\ x &= -\frac {\left (-c_1 \,t^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 t} \\ x &= \frac {\left (-c_1 \,t^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 23

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

\begin{align*} x(t)\to 0 \\ x(t)\to \frac {c_1}{\sqrt [3]{t}} \\ x(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.217 (sec). Leaf size: 8

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

\[ x{\left (t \right )} = \frac {C_{1}}{\sqrt [3]{t}} \]