76.26.11 problem 15

Internal problem ID [17764]
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 : 15
Date solved : Thursday, March 13, 2025 at 10:48:49 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+6 x_{2} \left (t \right )+2 x_{3} \left (t \right )-2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-3 x_{2} \left (t \right )-6 x_{3} \left (t \right )+2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-4 x_{1} \left (t \right )+8 x_{2} \left (t \right )+3 x_{3} \left (t \right )-4 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=2 x_{1} \left (t \right )-2 x_{2} \left (t \right )-6 x_{3} \left (t \right )+x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.140 (sec). Leaf size: 121

ode:=[diff(x__1(t),t) = -3*x__1(t)+6*x__2(t)+2*x__3(t)-2*x__4(t), diff(x__2(t),t) = 2*x__1(t)-3*x__2(t)-6*x__3(t)+2*x__4(t), diff(x__3(t),t) = -4*x__1(t)+8*x__2(t)+3*x__3(t)-4*x__4(t), diff(x__4(t),t) = 2*x__1(t)-2*x__2(t)-6*x__3(t)+x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{t}+c_{3} {\mathrm e}^{-t} \sin \left (4 t \right )+c_4 \,{\mathrm e}^{-t} \cos \left (4 t \right ) \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{t}+c_{3} {\mathrm e}^{-t} \cos \left (4 t \right )-c_4 \,{\mathrm e}^{-t} \sin \left (4 t \right ) \\ x_{3} \left (t \right ) &= {\mathrm e}^{-t} \left (c_4 \cos \left (4 t \right )+c_{3} \sin \left (4 t \right )\right ) \\ x_{4} \left (t \right ) &= -{\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{t}+c_{3} {\mathrm e}^{-t} \cos \left (4 t \right )-c_4 \,{\mathrm e}^{-t} \sin \left (4 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 247

ode={D[x1[t],t]==-3*x1[t]+6*x2[t]+2*x3[t]-2*x4[t],D[x2[t],t]==2*x1[t]-3*x2[t]-6*x3[t]+2*x4[t],D[x3[t],t]==-4*x1[t]+8*x2[t]+3*x3[t]-4*x4[t],D[x4[t],t]==2*x1[t]-2*x2[t]-6*x3[t]+1*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left (c_1 e^{2 t}-c_2 e^{2 t}-c_3 e^{2 t}+c_4 e^{2 t}+c_3 \cos (4 t)-(c_1-2 c_2-c_3+c_4) \sin (4 t)+c_2-c_4\right ) \\ \text {x2}(t)\to e^{-t} \left ((c_1-c_2-c_3+c_4) e^{2 t}-(c_1-2 c_2-c_3+c_4) \cos (4 t)-c_3 \sin (4 t)\right ) \\ \text {x3}(t)\to e^{-t} (c_3 \cos (4 t)-(c_1-2 c_2-c_3+c_4) \sin (4 t)) \\ \text {x4}(t)\to e^{-t} \left (c_1 e^{2 t}-c_2 e^{2 t}-c_3 e^{2 t}+c_4 e^{2 t}-(c_1-2 c_2-c_3+c_4) \cos (4 t)-c_3 \sin (4 t)-c_2+c_4\right ) \\ \end{align*}

✓ Sympy. Time used: 0.203 (sec). Leaf size: 114

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
x__4 = Function("x__4") 
ode=[Eq(3*x__1(t) - 6*x__2(t) - 2*x__3(t) + 2*x__4(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + 3*x__2(t) + 6*x__3(t) - 2*x__4(t) + Derivative(x__2(t), t),0),Eq(4*x__1(t) - 8*x__2(t) - 3*x__3(t) + 4*x__4(t) + Derivative(x__3(t), t),0),Eq(-2*x__1(t) + 2*x__2(t) + 6*x__3(t) - x__4(t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

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