30.2.28 problem 40

Internal problem ID [7456]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.3, Linear equations. Exercises. page 54
Problem number : 40
Date solved : Tuesday, September 30, 2025 at 04:36:21 PM
CAS classification : [_quadrature]

\begin{align*} u^{\prime }&=\alpha \left (1-u\right )-\beta u \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 21

ode:=diff(u(t),t) = alpha*(1-u(t))-beta*u(t); 
dsolve(ode,u(t), singsol=all);
 

\[ u = {\mathrm e}^{-\left (\alpha +\beta \right ) t} c_1 +\frac {\alpha }{\alpha +\beta } \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 35

ode=D[u[t],t]==\[Alpha]*(1-u[t])-\[Beta]*u[t]; 
ic={}; 
DSolve[{ode,ic},u[t],t,IncludeSingularSolutions->True]
 

\begin{align*} u(t)&\to \frac {\alpha }{\alpha +\beta }+c_1 e^{-t (\alpha +\beta )}\\ u(t)&\to \frac {\alpha }{\alpha +\beta } \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
u = Function("u") 
ode = Eq(-Alpha*(1 - u(t)) + BETA*u(t) + Derivative(u(t), t),0) 
ics = {} 
dsolve(ode,func=u(t),ics=ics)
 

\[ u{\left (t \right )} = \frac {\mathrm {A}}{\mathrm {A} + \beta } + C_{1} e^{- t \left (\mathrm {A} + \beta \right )} \]