76.26.13 problem 17

Internal problem ID [17774]
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 : 17
Date solved : Monday, March 31, 2025 at 04:27:21 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )-5 x_{2} \left (t \right )+8 x_{3} \left (t \right )+14 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-6 x_{1} \left (t \right )-8 x_{2} \left (t \right )+11 x_{3} \left (t \right )+27 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-6 x_{1} \left (t \right )-4 x_{2} \left (t \right )+7 x_{3} \left (t \right )+17 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 )+4 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.670 (sec). Leaf size: 145

ode:=[diff(x__1(t),t) = -3*x__1(t)-5*x__2(t)+8*x__3(t)+14*x__4(t), diff(x__2(t),t) = -6*x__1(t)-8*x__2(t)+11*x__3(t)+27*x__4(t), diff(x__3(t),t) = -6*x__1(t)-4*x__2(t)+7*x__3(t)+17*x__4(t), diff(x__4(t),t) = -2*x__2(t)+2*x__3(t)+4*x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 274

ode={D[x1[t],t]==-3*x1[t]-5*x2[t]+8*x3[t]+14*x4[t],D[x2[t],t]==-6*x1[t]-8*x2[t]+11*x3[t]+27*x4[t],D[x3[t],t]==-6*x1[t]-4*x2[t]+7*x3[t]+17*x4[t],D[x4[t],t]==0*x1[t]-2*x2[t]+2*x3[t]+4*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.238 (sec). Leaf size: 119

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) + 5*x__2(t) - 8*x__3(t) - 14*x__4(t) + Derivative(x__1(t), t),0),Eq(6*x__1(t) + 8*x__2(t) - 11*x__3(t) - 27*x__4(t) + Derivative(x__2(t), t),0),Eq(6*x__1(t) + 4*x__2(t) - 7*x__3(t) - 17*x__4(t) + Derivative(x__3(t), t),0),Eq(2*x__2(t) - 2*x__3(t) - 4*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)
 

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