80.2.4 problem 5

Internal problem ID [18543]
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 : Tuesday, January 28, 2025 at 11:55:16 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

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

DSolve[D[x[t],t]==k*(A-n*x[t])*(M-m*x[t]),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*}