65.3.11 problem 8.7

Internal problem ID [13729]
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 : Tuesday, January 28, 2025 at 05:59:55 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*}

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 22

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

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

✓ Solution by Mathematica

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

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

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