78.5.11 problem 5.e

Internal problem ID [21050]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 5.e
Date solved : Thursday, October 02, 2025 at 07:01:46 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.104 (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: 0.121 (sec). Leaf size: 76

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)
 

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