65.3.1 problem 8.1 (i)

Internal problem ID [13640]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 8, Separable equations. Exercises page 72
Problem number : 8.1 (i)
Date solved : Wednesday, March 05, 2025 at 10:06:05 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} x \left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 14

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

\[ x \left (t \right ) = 1+2 \,{\mathrm e}^{-\frac {t^{4}}{4}} \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 18

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

\[ x(t)\to 2 e^{-\frac {t^4}{4}}+1 \]

✓ Sympy. Time used: 0.305 (sec). Leaf size: 12

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

\[ x{\left (t \right )} = 1 + 2 e^{- \frac {t^{4}}{4}} \]