80.2.4 problem 5

Internal problem ID [18472]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 24. Problems at page 62
Problem number : 5
Date solved : Monday, March 31, 2025 at 05:30:59 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=k \left (A -n x\right ) \left (M -m x\right ) \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 47

ode:=diff(x(t),t) = k*(A-n*x(t))*(M-m*x(t)); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {-A \,{\mathrm e}^{-k \left (c_1 +t \right ) \left (A m -M n \right )}+M}{-{\mathrm e}^{-k \left (c_1 +t \right ) \left (A m -M n \right )} n +m} \]

✓ Mathematica. Time used: 2.839 (sec). Leaf size: 82

ode=D[x[t],t]==k*(A-n*x[t])*(M-m*x[t]); 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {A e^{M n (k t+c_1)}-M e^{A m (k t+c_1)}}{n e^{M n (k t+c_1)}-m e^{A m (k t+c_1)}} \\ x(t)\to \frac {M}{m} \\ x(t)\to \frac {A}{n} \\ \end{align*}

✓ Sympy. Time used: 1.349 (sec). Leaf size: 49

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

\[ x{\left (t \right )} = \frac {- \frac {A e^{M n \left (C_{1} + k t\right )}}{n} + \frac {M e^{A m \left (C_{1} + k t\right )}}{m}}{e^{A m \left (C_{1} + k t\right )} - e^{M n \left (C_{1} + k t\right )}} \]