76.26.12 problem 16

Internal problem ID [17773]
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 : 16
Date solved : Monday, March 31, 2025 at 04:27:18 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )-4 x_{2} \left (t \right )+5 x_{3} \left (t \right )+9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-5 x_{2} \left (t \right )+4 x_{3} \left (t \right )+12 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )-x_{3} \left (t \right )+2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-2 x_{2} \left (t \right )+2 x_{3} \left (t \right )+3 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.161 (sec). Leaf size: 209

ode:=[diff(x__1(t),t) = -3*x__1(t)-4*x__2(t)+5*x__3(t)+9*x__4(t), diff(x__2(t),t) = -2*x__1(t)-5*x__2(t)+4*x__3(t)+12*x__4(t), diff(x__3(t),t) = -2*x__1(t)-x__3(t)+2*x__4(t), diff(x__4(t),t) = -2*x__2(t)+2*x__3(t)+3*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -c_1 \,{\mathrm e}^{-t} \cos \left (2 t \right )+c_2 \,{\mathrm e}^{-t} \sin \left (2 t \right )-\frac {c_3 \,{\mathrm e}^{-2 t} \cos \left (t \right )}{2}+\frac {c_4 \,{\mathrm e}^{-2 t} \sin \left (t \right )}{2}-c_1 \,{\mathrm e}^{-t} \sin \left (2 t \right )-c_2 \,{\mathrm e}^{-t} \cos \left (2 t \right )+\frac {c_3 \,{\mathrm e}^{-2 t} \sin \left (t \right )}{2}+\frac {c_4 \,{\mathrm e}^{-2 t} \cos \left (t \right )}{2} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{-t} \cos \left (2 t \right )-c_2 \,{\mathrm e}^{-t} \sin \left (2 t \right )-c_1 \,{\mathrm e}^{-t} \sin \left (2 t \right )-c_2 \,{\mathrm e}^{-t} \cos \left (2 t \right )+c_3 \,{\mathrm e}^{-2 t} \sin \left (t \right )+c_4 \,{\mathrm e}^{-2 t} \cos \left (t \right ) \\ x_{3} \left (t \right ) &= c_1 \,{\mathrm e}^{-t} \sin \left (2 t \right )+c_2 \,{\mathrm e}^{-t} \cos \left (2 t \right )+c_3 \,{\mathrm e}^{-2 t} \sin \left (t \right )+c_4 \,{\mathrm e}^{-2 t} \cos \left (t \right ) \\ x_{4} \left (t \right ) &= -{\mathrm e}^{-t} \left (\sin \left (2 t \right ) c_1 +\cos \left (2 t \right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 258

ode={D[x1[t],t]==-3*x1[t]-4*x2[t]+5*x3[t]+9*x4[t],D[x2[t],t]==-2*x1[t]-5*x2[t]+4*x3[t]+12*x4[t],D[x3[t],t]==-2*x1[t]+0*x2[t]-1*x3[t]+2*x4[t],D[x4[t],t]==0*x1[t]-2*x2[t]+2*x3[t]+3*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-2 t} \left ((c_1+c_2-c_3-3 c_4) \cos (t)-(c_2-c_3-3 c_4) e^t \cos (2 t)-\sin (t) \left (2 (c_2-c_3-c_4) e^t \cos (t)+c_1+c_2-2 c_3-4 c_4\right )\right ) \\ \text {x2}(t)\to e^{-2 t} \left ((c_2-c_3-c_4) e^t \cos (2 t)+(-2 c_1-2 c_2+3 c_3+7 c_4) \sin (t)+\cos (t) \left (-2 (c_2-c_3-3 c_4) e^t \sin (t)+c_3+c_4\right )\right ) \\ \text {x3}(t)\to e^{-2 t} \left (c_4 \left (-e^t\right ) \cos (2 t)+(-2 c_1-2 c_2+3 c_3+7 c_4) \sin (t)+\cos (t) \left (2 (c_2-c_3-2 c_4) e^t \sin (t)+c_3+c_4\right )\right ) \\ \text {x4}(t)\to e^{-t} (c_4 \cos (2 t)+(-c_2+c_3+2 c_4) \sin (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.290 (sec). Leaf size: 167

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) + 4*x__2(t) - 5*x__3(t) - 9*x__4(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) + 5*x__2(t) - 4*x__3(t) - 12*x__4(t) + Derivative(x__2(t), t),0),Eq(2*x__1(t) + x__3(t) - 2*x__4(t) + Derivative(x__3(t), t),0),Eq(2*x__2(t) - 2*x__3(t) - 3*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)
 

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