80.7.5 problem A 5

Internal problem ID [21324]
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 5
Date solved : Thursday, October 02, 2025 at 07:28:26 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 36

ode:=[diff(x(t),t) = 2*x(t)+y(t), diff(y(t),t) = -x(t)+2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (c_2 \cos \left (t \right )+c_1 \sin \left (t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{2 t} \left (\cos \left (t \right ) c_1 -\sin \left (t \right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 43

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

\begin{align*} x(t)&\to e^{2 t} (c_1 \cos (t)+c_2 \sin (t))\\ y(t)&\to e^{2 t} (c_2 \cos (t)-c_1 \sin (t)) \end{align*}

✓ Sympy. Time used: 0.061 (sec). Leaf size: 44

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

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