87.31.16 problem 16

Internal problem ID [23932]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 321
Problem number : 16
Date solved : Sunday, October 12, 2025 at 05:55:17 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+x \left (t \right )^{2}\\ \frac {d}{d t}y \left (t \right )&=-3 y \left (t \right )+x \left (t \right ) y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 26

ode:=[diff(x(t),t) = -x(t)+x(t)^2, diff(y(t),t) = -3*y(t)+x(t)*y(t)]; 
dsolve(ode);
 

\begin{align*} \left \{x \left (t \right ) &= \frac {1}{1+{\mathrm e}^{t} c_2}\right \} \\ \{y \left (t \right ) &= c_1 \,{\mathrm e}^{\int \left (x \left (t \right )-3\right )d t}\} \\ \end{align*}

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 39

ode={D[x[t],t]==-x[t]+x[t]^2,D[y[t],t]==-3*y[t]+x[t]*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{1+e^{t+c_1}}\\ y(t)&\to \frac {c_2 e^{-2 (t+c_1)}}{1+e^{t+c_1}} \end{align*}

✓ Sympy. Time used: 0.276 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t)**2 + x(t) + Derivative(x(t), t),0),Eq(-x(t)*y(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {1}{1 - e^{C_{2} + t}}, \ y{\left (t \right )} = \frac {C_{1}}{\left (1 - e^{C_{2} + t}\right )^{3} \left (1 - \frac {2}{1 - e^{C_{2} + t}} + \frac {1}{\left (1 - e^{C_{2} + t}\right )^{2}}\right )}\right ] \]