78.6.1 problem 1

Internal problem ID [21064]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 7, Nonlinear systems. Problems section 7.11
Problem number : 1
Date solved : Thursday, October 02, 2025 at 07:01:55 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=k y-c y^{2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=y_{0} \\ \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 28

ode:=diff(y(t),t) = k*y(t)-c*y(t)^2; 
ic:=[y(0) = y__0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {k y_{0}}{\left (y_{0} c -k \right ) {\mathrm e}^{-k t}-y_{0} c} \]

✓ Mathematica. Time used: 0.43 (sec). Leaf size: 27

ode=D[y[t],t]==k*y[t]-c*y[t]^2; 
ic={y[0]==y0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {k \text {y0} e^{k t}}{c \text {y0} \left (e^{k t}-1\right )+k} \end{align*}

✓ Sympy. Time used: 0.294 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y0 = symbols("y0") 
y = Function("y") 
ode = Eq(-c*y(t)**2 - k*y(t) + Derivative(y(t), t),0) 
ics = {y(0): y0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {k e^{k \left (t + \frac {\log {\left (\frac {c y_{0}}{c y_{0} + k} \right )}}{k}\right )}}{c \left (1 - e^{k \left (t + \frac {\log {\left (\frac {c y_{0}}{c y_{0} + k} \right )}}{k}\right )}\right )} \]