80.7.7 problem A 7

Internal problem ID [21326]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 7. System of first order equations. Excercise 7.6 at page 162
Problem number : A 7
Date solved : Thursday, October 02, 2025 at 07:28:27 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 18

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

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

✓ Sympy. Time used: 0.029 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) + Derivative(x(t), t),0),Eq(-y(t) + Derivative(y(t), t),0)] 
ics = {x(0): 1, y(0): -1} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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