87.29.8 problem 8

Internal problem ID [23888]
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 304
Problem number : 8
Date solved : Thursday, October 02, 2025 at 09:46:18 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*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 18

ode:=[diff(x(t),t) = -x(t), diff(y(t),t) = -x(t)-y(t)]; 
ic:=[x(0) = 1, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 21

ode={D[x[t],t]==-x[t],D[y[t],t]==-x[t]-y[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.048 (sec). Leaf size: 15

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 = {x(0): 1, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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