72.27.1 problem 5 (a,c)

Internal problem ID [19752]
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 : 5 (a,c)
Date solved : Thursday, October 02, 2025 at 04:41:59 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+3 y\\ y^{\prime }&=3 x \left (t \right )+y \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=5 \\ y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.142 (sec). Leaf size: 33

ode:=[diff(x(t),t) = x(t)+3*y(t), diff(y(t),t) = 3*x(t)+y(t)]; 
ic:=[x(0) = 5, y(0) = 1]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

ode={D[x[t],t]==x[t]+3*y[t],D[y[t],t]==3*x[t]+y[t]}; 
ic={x[0]==5,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-2 t} \left (3 e^{6 t}+2\right )\\ y(t)&\to e^{-2 t} \left (3 e^{6 t}-2\right ) \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 31

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

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