80.7.3 problem A 3

Internal problem ID [21322]
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 3
Date solved : Thursday, October 02, 2025 at 07:28:25 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.058 (sec). Leaf size: 23

ode:=[diff(x(t),t) = a*x(t)+y(t), diff(y(t),t) = a*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 29

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

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

✓ Sympy. Time used: 0.044 (sec). Leaf size: 26

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

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