78.27.2 problem 6 (a)

Internal problem ID [18388]
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 55. Linear systems. Problems at page 496
Problem number : 6 (a)
Date solved : Thursday, March 13, 2025 at 11:55:25 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 35

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

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+{\mathrm e}^{-t} c_{2} \\ y &= \frac {3 c_{1} {\mathrm e}^{4 t}}{2}-{\mathrm e}^{-t} c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 74

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

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

✓ Sympy. Time used: 0.091 (sec). Leaf size: 31

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

\[ \left [ x{\left (t \right )} = - C_{1} e^{- t} + \frac {2 C_{2} e^{4 t}}{3}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{4 t}\right ] \]