80.3.6 problem 6

Internal problem ID [21170]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 3. First order nonlinear differential equations. Excercise 3.7 at page 67
Problem number : 6
Date solved : Thursday, October 02, 2025 at 07:15:18 PM
CAS classification : [_separable]

\begin{align*} x^{\prime }&=4 t^{3} x^{4} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 57

ode:=diff(x(t),t) = 4*t^3*x(t)^4; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x &= \frac {1}{\left (-3 t^{4}+c_1 \right )^{{1}/{3}}} \\ x &= -\frac {1+i \sqrt {3}}{2 \left (-3 t^{4}+c_1 \right )^{{1}/{3}}} \\ x &= \frac {i \sqrt {3}-1}{2 \left (-3 t^{4}+c_1 \right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 86

ode=D[x[t],t]==4*t^3*x[t]^4; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {\sqrt [3]{-\frac {1}{3}}}{\sqrt [3]{-t^4-c_1}}\\ x(t)&\to \frac {1}{\sqrt [3]{3} \sqrt [3]{-t^4-c_1}}\\ x(t)&\to \frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{-t^4-c_1}}\\ x(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.710 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-4*t**3*x(t)**4 + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ \left [ x{\left (t \right )} = \sqrt [3]{- \frac {1}{C_{1} + 3 t^{4}}}, \ x{\left (t \right )} = \frac {\sqrt [3]{- \frac {1}{C_{1} + t^{4}}} \left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right )}{6}, \ x{\left (t \right )} = \frac {\sqrt [3]{- \frac {1}{C_{1} + t^{4}}} \left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right )}{6}\right ] \]