65.3.11 problem 8.7

Internal problem ID [13658]
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 : 8.7
Date solved : Monday, March 31, 2025 at 08:04:19 AM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=k x-x^{2} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 22

ode:=diff(x(t),t) = k*x(t)-x(t)^2; 
ic:=x(0) = x__0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \frac {k x_{0}}{\left (-x_{0} +k \right ) {\mathrm e}^{-k t}+x_{0}} \]

✓ Mathematica. Time used: 0.267 (sec). Leaf size: 50

ode=D[x[t],t]==k*x[t]-x[t]^2; 
ic={x[0]==x0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \text {InverseFunction}\left [\int _0^{\text {$\#$1}}\frac {1}{(k-K[1]) K[1]}dK[1]\&\right ]\left [\int _0^{\text {x0}}\frac {1}{k K[1]-K[1]^2}dK[1]+t\right ] \]

✓ Sympy. Time used: 0.397 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
k = symbols("k") 
x = Function("x") 
ode = Eq(-k*x(t) + x(t)**2 + Derivative(x(t), t),0) 
ics = {x(0): x__0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {k e^{k \left (t + \frac {\log {\left (- \frac {x^{0}}{k - x^{0}} \right )}}{k}\right )}}{e^{k \left (t + \frac {\log {\left (- \frac {x^{0}}{k - x^{0}} \right )}}{k}\right )} - 1} \]