80.7.21 problem B 13

Internal problem ID [21340]
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 : B 13
Date solved : Thursday, October 02, 2025 at 07:28:34 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=x+y \left (t \right )\\ 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 )&=2 \\ \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 19

ode:=[diff(x(t),t) = x(t)+y(t), diff(y(t),t) = -y(t)]; 
ic:=[x(0) = -1, y(0) = 2]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -{\mathrm e}^{-t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 22

ode={D[x[t],t]==x[t]+y[t],D[y[t],t]==-y[t]}; 
ic={x[0]==-1,y[0]==2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -e^{-t}\\ y(t)&\to 2 e^{-t} \end{align*}

✓ Sympy. Time used: 0.041 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - y(t) + Derivative(x(t), t),0),Eq(y(t) + Derivative(y(t), t),0)] 
ics = {x(0): -1, y(0): 2} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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