28.4.3 problem 7.3

Internal problem ID [4535]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.3
Date solved : Sunday, March 30, 2025 at 03:25:41 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-x \left (t \right )+3 y \left (t \right )&=0\\ 3 x \left (t \right )-\frac {d}{d t}y \left (t \right )+y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 41

ode:=[diff(x(t),t)-x(t)+3*y(t) = 0, 3*x(t)-diff(y(t),t)+y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_2 \cos \left (3 t \right )+c_1 \sin \left (3 t \right )\right ) \\ y \left (t \right ) &= -{\mathrm e}^{t} \left (\cos \left (3 t \right ) c_1 -\sin \left (3 t \right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 47

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

\begin{align*} x(t)\to e^t (c_1 \cos (3 t)-c_2 \sin (3 t)) \\ y(t)\to e^t (c_2 \cos (3 t)+c_1 \sin (3 t)) \\ \end{align*}

✓ Sympy. Time used: 0.097 (sec). Leaf size: 46

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)
 

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