14.2.11 problem 11

Internal problem ID [2499]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.4 separable equations. Excercises page 24
Problem number : 11
Date solved : Monday, January 27, 2025 at 05:54:37 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=k \left (a -y\right ) \left (b -y\right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.233 (sec). Leaf size: 35

dsolve([diff(y(t),t)=k*(a-y(t))*(b-y(t)),y(0) = 0],y(t), singsol=all)
 

\[ y = \frac {a b \left ({\mathrm e}^{k t \left (a -b \right )}-1\right )}{{\mathrm e}^{k t \left (a -b \right )} a -b} \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 43

DSolve[{D[y[t],t]==k*(a-y[t])*(b-y[t]),{y[0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {a b \left (e^{a k t}-e^{b k t}\right )}{a e^{a k t}-b e^{b k t}} \]