7.3.11 problem 11

Internal problem ID [2328]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.4. Page 24
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 05:26:56 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*}

✓ Maple. Time used: 0.259 (sec). Leaf size: 35

ode:=diff(y(t),t) = k*(a-y(t))*(b-y(t)); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 43

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

\begin{align*} 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}} \end{align*}

✓ Sympy. Time used: 0.768 (sec). Leaf size: 66

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

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