76.27.6 problem 6

Internal problem ID [17781]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.5 (Fundamental Matrices and the Exponential of a Matrix). Problems at page 430
Problem number : 6
Date solved : Monday, March 31, 2025 at 04:27:33 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.131 (sec). Leaf size: 45

ode:=[diff(x__1(t),t) = -x__1(t)-4*x__2(t), diff(x__2(t),t) = x__1(t)-x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_2 \cos \left (2 t \right )+c_1 \sin \left (2 t \right )\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-t} \left (\cos \left (2 t \right ) c_1 -\sin \left (2 t \right ) c_2 \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 55

ode={D[x1[t],t]==-1*x1[t]-4*x2[t],D[x2[t],t]==1*x1[t]-1*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t} (c_1 \cos (2 t)-2 c_2 \sin (2 t)) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} (2 c_2 \cos (2 t)+c_1 \sin (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__1(t) + 4*x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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