78.29.5 problem 1 (e)

Internal problem ID [18413]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 10. Systems of First Order Equations. Section 60. Critical Points and Stability for Linear Systems. Problems at page 539
Problem number : 1 (e)
Date solved : Monday, March 31, 2025 at 05:27:40 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 29

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

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

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

ode={D[x[t],t]==-4*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^{-3 t} (c_1 (-t)-c_2 t+c_1) \\ y(t)\to e^{-3 t} ((c_1+c_2) t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.084 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(4*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)
 

\[ \left [ x{\left (t \right )} = - C_{2} t e^{- 3 t} - \left (C_{1} - C_{2}\right ) e^{- 3 t}, \ y{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} t e^{- 3 t}\right ] \]