28.4.31 problem 7.31

Internal problem ID [4563]
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 : 7.31
Date solved : Tuesday, March 04, 2025 at 06:52:28 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-3 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )+x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 40

ode:=[diff(x__1(t),t) = x__1(t)-3*x__2(t), diff(x__2(t),t) = 3*x__1(t)+x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (\sin \left (3 t \right ) c_{1} +\cos \left (3 t \right ) c_{2} \right ) \\ x_{2} \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[x1[t],t]==x1[t]-3*x2[t],D[x2[t],t]==3*x1[t]+x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^t (c_1 \cos (3 t)-c_2 \sin (3 t)) \\ \text {x2}(t)\to e^t (c_2 \cos (3 t)+c_1 \sin (3 t)) \\ \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) + 3*x__2(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) - x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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