63.18.2 problem 2(b)

Internal problem ID [13129]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 190
Problem number : 2(b)
Date solved : Wednesday, March 05, 2025 at 09:17:56 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 111

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

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

✓ Sympy. Time used: 0.181 (sec). Leaf size: 65

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

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