63.4.19 problem 9

Internal problem ID [12983]
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 1.3.1 Separable equations. Exercises page 26
Problem number : 9
Date solved : Wednesday, March 05, 2025 at 08:55:57 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=x \left (4+x\right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 16

ode:=diff(x(t),t) = x(t)*(4+x(t)); 
ic:=x(0) = 1; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {4}{-1+5 \,{\mathrm e}^{-4 t}} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 21

ode=D[x[t],t]==x[t]*(4+x[t]); 
ic={x[0]==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to -\frac {4 e^{4 t}}{e^{4 t}-5} \]

✓ Sympy. Time used: 0.390 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-(x(t) + 4)*x(t) + Derivative(x(t), t),0) 
ics = {x(0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {4}{-1 + 5 e^{- 4 t}} \]