63.19.4 problem 1(d)

Internal problem ID [13139]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 202
Problem number : 1(d)
Date solved : Wednesday, March 05, 2025 at 09:18:06 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 27

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

\begin{align*} x \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-4 t}}{2}+c_{1} {\mathrm e}^{-2 t} \\ y &= c_{2} {\mathrm e}^{-4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 43

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

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

✓ Sympy. Time used: 0.092 (sec). Leaf size: 26

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(4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} e^{- 4 t}}{2} + C_{2} e^{- 2 t}, \ y{\left (t \right )} = C_{1} e^{- 4 t}\right ] \]