63.19.2 problem 1(b)

Internal problem ID [13137]
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(b)
Date solved : Wednesday, March 05, 2025 at 09:18:04 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 35

ode:=[diff(x(t),t) = -3*y(t), diff(y(t),t) = -2*x(t)+y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+{\mathrm e}^{-2 t} c_{2} \\ y &= -c_{1} {\mathrm e}^{3 t}+\frac {2 \,{\mathrm e}^{-2 t} c_{2}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 74

ode={D[x[t],t]==-3*y[t],D[y[t],t]==-2*x[t]+y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-2 t} \left (c_1 \left (2 e^{5 t}+3\right )-3 c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} e^{-2 t} \left (c_2 \left (3 e^{5 t}+2\right )-2 c_1 \left (e^{5 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*y(t) + Derivative(x(t), t),0),Eq(2*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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