76.5.20 problem 20

Internal problem ID [17369]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 20
Date solved : Monday, March 31, 2025 at 04:09:22 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

ode:=diff(y(t),t) = r*y(t)-k^2*y(t)^2; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {r}{{\mathrm e}^{-r t} c_1 r +k^{2}} \]

✓ Mathematica. Time used: 0.254 (sec). Leaf size: 49

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

\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (r-k^2 K[1]\right )}dK[1]\&\right ][t+c_1] \\ y(t)\to 0 \\ y(t)\to \frac {r}{k^2} \\ \end{align*}

✓ Sympy. Time used: 0.408 (sec). Leaf size: 22

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

\[ y{\left (t \right )} = \frac {r e^{r \left (C_{1} + t\right )}}{k^{2} \left (e^{r \left (C_{1} + t\right )} - 1\right )} \]