65.5.10 problem 10.5

Internal problem ID [13749]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 10, Two tricks for nonlinear equations. Exercises page 97
Problem number : 10.5
Date solved : Tuesday, January 28, 2025 at 06:01:46 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 18

dsolve(diff(x(t),t)=k*x(t)-x(t)^2,x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.272 (sec). Leaf size: 42

DSolve[D[x[t],t]==k*x[t]-x[t]^2,x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(k-K[1]) K[1]}dK[1]\&\right ][t+c_1] \\ x(t)\to 0 \\ x(t)\to k \\ \end{align*}