76.26.10 problem 10

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 : 10
Date solved : Thursday, March 13, 2025 at 10:48:44 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right )+4 x_{3} \left (t \right )+2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-19 x_{1} \left (t \right )-6 x_{2} \left (t \right )+6 x_{3} \left (t \right )+16 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-9 x_{1} \left (t \right )-x_{2} \left (t \right )+x_{3} \left (t \right )+6 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-5 x_{1} \left (t \right )-3 x_{2} \left (t \right )+6 x_{3} \left (t \right )+5 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.148 (sec). Leaf size: 196

ode:=[diff(x__1(t),t) = -2*x__1(t)-x__2(t)+4*x__3(t)+2*x__4(t), diff(x__2(t),t) = -19*x__1(t)-6*x__2(t)+6*x__3(t)+16*x__4(t), diff(x__3(t),t) = -9*x__1(t)-x__2(t)+x__3(t)+6*x__4(t), diff(x__4(t),t) = -5*x__1(t)-3*x__2(t)+6*x__3(t)+5*x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 470

ode={D[x1[t],t]==-2*x1[t]-1*x2[t]+4*x3[t]+2*x4[t],D[x2[t],t]==-19*x1[t]-6*x2[t]+6*x3[t]+16*x4[t],D[x3[t],t]==-9*x1[t]-1*x2[t]+1*x3[t]+6*x4[t],D[x4[t],t]==-5*x1[t]-3*x2[t]+6*x3[t]+5*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 ((-3 c_1-c_2+c_3+3 c_4) \cos (2 t)+(4 c_1+c_2-c_3-3 c_4) e^t \cos (3 t)-c_1 \sin (2 t)-c_2 \sin (2 t)+c_3 \sin (2 t)+c_4 \sin (2 t)-c_1 e^t \sin (3 t)+c_3 e^t \sin (3 t)+c_4 e^t \sin (3 t)\right ) \\ \text {x2}(t)\to e^{-t} \left ((c_1+c_2-c_3-c_4) \cos (2 t)-(c_1-c_3-c_4) e^t \cos (3 t)-3 c_1 \sin (2 t)-c_2 \sin (2 t)+c_3 \sin (2 t)+3 c_4 \sin (2 t)-4 c_1 e^t \sin (3 t)-c_2 e^t \sin (3 t)+c_3 e^t \sin (3 t)+3 c_4 e^t \sin (3 t)\right ) \\ \text {x3}(t)\to e^{-t} \left ((c_1-c_4) \cos (2 t)-(c_1-c_3-c_4) e^t \cos (3 t)+2 c_1 \sin (2 t)+c_2 \sin (2 t)-c_3 \sin (2 t)-2 c_4 \sin (2 t)-4 c_1 e^t \sin (3 t)-c_2 e^t \sin (3 t)+c_3 e^t \sin (3 t)+3 c_4 e^t \sin (3 t)\right ) \\ \text {x4}(t)\to e^{-t} \left ((-4 c_1-c_2+c_3+4 c_4) \cos (2 t)+(4 c_1+c_2-c_3-3 c_4) e^t \cos (3 t)-3 c_1 \sin (2 t)-2 c_2 \sin (2 t)+2 c_3 \sin (2 t)+3 c_4 \sin (2 t)-c_1 e^t \sin (3 t)+c_3 e^t \sin (3 t)+c_4 e^t \sin (3 t)\right ) \\ \end{align*}

✓ Sympy. Time used: 0.309 (sec). Leaf size: 184

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(2*x__1(t) + x__2(t) - 4*x__3(t) - 2*x__4(t) + Derivative(x__1(t), t),0),Eq(19*x__1(t) + 6*x__2(t) - 6*x__3(t) - 16*x__4(t) + Derivative(x__2(t), t),0),Eq(9*x__1(t) + x__2(t) - x__3(t) - 6*x__4(t) + Derivative(x__3(t), t),0),Eq(5*x__1(t) + 3*x__2(t) - 6*x__3(t) - 5*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)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{3} \sin {\left (3 t \right )} + C_{4} \cos {\left (3 t \right )} - \left (\frac {C_{1}}{5} - \frac {3 C_{2}}{5}\right ) e^{- t} \cos {\left (2 t \right )} - \left (\frac {3 C_{1}}{5} + \frac {C_{2}}{5}\right ) e^{- t} \sin {\left (2 t \right )}, \ x^{2}{\left (t \right )} = - C_{3} \cos {\left (3 t \right )} - C_{4} \sin {\left (3 t \right )} - \left (\frac {C_{1}}{5} - \frac {3 C_{2}}{5}\right ) e^{- t} \sin {\left (2 t \right )} + \left (\frac {3 C_{1}}{5} + \frac {C_{2}}{5}\right ) e^{- t} \cos {\left (2 t \right )}, \ x^{3}{\left (t \right )} = - C_{3} \cos {\left (3 t \right )} - C_{4} \sin {\left (3 t \right )} - \left (\frac {C_{1}}{5} + \frac {2 C_{2}}{5}\right ) e^{- t} \cos {\left (2 t \right )} + \left (\frac {2 C_{1}}{5} - \frac {C_{2}}{5}\right ) e^{- t} \sin {\left (2 t \right )}, \ x^{4}{\left (t \right )} = - C_{1} e^{- t} \sin {\left (2 t \right )} + C_{2} e^{- t} \cos {\left (2 t \right )} - C_{3} \sin {\left (3 t \right )} + C_{4} \cos {\left (3 t \right )}\right ] \]