80.11.1 problem 1

Internal problem ID [21409]
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 12. Stability theory. Excercise 12.6 at page 270
Problem number : 1
Date solved : Thursday, October 02, 2025 at 07:30:54 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-2 x+y \left (t \right )\\ y^{\prime }\left (t \right )&=7 x-4 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 94

ode:=[diff(x(t),t) = -2*x(t)+y(t), diff(y(t),t) = 7*x(t)-4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\left (-3+2 \sqrt {2}\right ) t}+c_2 \,{\mathrm e}^{-\left (3+2 \sqrt {2}\right ) t} \\ y \left (t \right ) &= 2 c_1 \,{\mathrm e}^{\left (-3+2 \sqrt {2}\right ) t} \sqrt {2}-2 c_2 \,{\mathrm e}^{-\left (3+2 \sqrt {2}\right ) t} \sqrt {2}-c_1 \,{\mathrm e}^{\left (-3+2 \sqrt {2}\right ) t}-c_2 \,{\mathrm e}^{-\left (3+2 \sqrt {2}\right ) t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 148

ode={D[x[t],t]==-2*x[t]+y[t],D[y[t],t]==7*x[t]-4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{8} e^{-\left (\left (3+2 \sqrt {2}\right ) t\right )} \left (c_1 \left (\left (4+\sqrt {2}\right ) e^{4 \sqrt {2} t}+4-\sqrt {2}\right )+\sqrt {2} c_2 \left (e^{4 \sqrt {2} t}-1\right )\right )\\ y(t)&\to \frac {1}{8} e^{-\left (\left (3+2 \sqrt {2}\right ) t\right )} \left (7 \sqrt {2} c_1 \left (e^{4 \sqrt {2} t}-1\right )-c_2 \left (\left (\sqrt {2}-4\right ) e^{4 \sqrt {2} t}-4-\sqrt {2}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode=[Eq(2*x(t) - y(t) + Derivative(x(t), t),0),Eq(-7*x(t) + 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

ValueError : 
Input to the funcs should be a list of functions.