76.26.2 problem 2

Internal problem ID [17763]
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 6.4 (Nondefective Matrices with Complex Eigenvalues). Problems at page 419
Problem number : 2
Date solved : Monday, March 31, 2025 at 04:26:59 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-4 x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )-2 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.147 (sec). Leaf size: 81

ode:=[diff(x__1(t),t) = 2*x__1(t)-4*x__2(t)-x__3(t), diff(x__2(t),t) = x__1(t)+x__2(t)+3*x__3(t), diff(x__3(t),t) = 3*x__1(t)-4*x__2(t)-2*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{t} \sin \left (4 t \right )+c_3 \,{\mathrm e}^{t} \cos \left (4 t \right ) \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}-c_2 \,{\mathrm e}^{t} \cos \left (4 t \right )+c_3 \,{\mathrm e}^{t} \sin \left (4 t \right ) \\ x_{3} \left (t \right ) &= -c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{t} \sin \left (4 t \right )+c_3 \,{\mathrm e}^{t} \cos \left (4 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 168

ode={D[x1[t],t]==2*x1[t]-4*x2[t]-1*x3[t],D[x2[t],t]==1*x1[t]+1*x2[t]+3*x3[t],D[x3[t],t]==3*x1[t]-4*x2[t]-2*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t} \left ((c_1+c_3) e^{2 t} \cos (4 t)+(c_1-2 c_2-c_3) e^{2 t} \sin (4 t)+c_1-c_3\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (-(c_1-2 c_2-c_3) e^{2 t} \cos (4 t)+(c_1+c_3) e^{2 t} \sin (4 t)+c_1-c_3\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{-t} \left ((c_1+c_3) e^{2 t} \cos (4 t)+(c_1-2 c_2-c_3) e^{2 t} \sin (4 t)-c_1+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 82

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-2*x__1(t) + 4*x__2(t) + x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - x__2(t) - 3*x__3(t) + Derivative(x__2(t), t),0),Eq(-3*x__1(t) + 4*x__2(t) + 2*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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