87.3.16 problem 20

Internal problem ID [23265]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 26
Problem number : 20
Date solved : Thursday, October 02, 2025 at 09:27:34 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 37

ode:=diff(x(t),t) = k*(a-x(t))*(b-x(t)); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {b \,{\mathrm e}^{\left (a -b \right ) \left (c_1 +t \right ) k}-a}{{\mathrm e}^{\left (a -b \right ) \left (c_1 +t \right ) k}-1} \]

✓ Mathematica. Time used: 1.05 (sec). Leaf size: 67

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

\begin{align*} x(t)&\to \frac {b e^{a (k t+c_1)}-a e^{b (k t+c_1)}}{e^{a (k t+c_1)}-e^{b (k t+c_1)}}\\ x(t)&\to a\\ x(t)&\to b \end{align*}

✓ Sympy. Time used: 0.835 (sec). Leaf size: 39

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

\[ x{\left (t \right )} = \frac {- a e^{b \left (C_{1} + k t\right )} + b e^{a \left (C_{1} + k t\right )}}{e^{a \left (C_{1} + k t\right )} - e^{b \left (C_{1} + k t\right )}} \]