87.31.26 problem 33

Internal problem ID [23942]
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 : 33
Date solved : Thursday, October 02, 2025 at 09:46:41 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 24

ode:=[diff(x(t),t) = -x(t), diff(y(t),t) = -x(t)-y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 30

ode={D[x[t],t]==-x[t],D[y[t],t]==-x[t]-y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.035 (sec). Leaf size: 22

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

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