76.26.5 problem 5

Internal problem ID [17758]
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 : 5
Date solved : Thursday, March 13, 2025 at 10:48:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-7 x_{1} \left (t \right )+6 x_{2} \left (t \right )-6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-9 x_{1} \left (t \right )+5 x_{2} \left (t \right )-9 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 85

ode:=[diff(x__1(t),t) = -7*x__1(t)+6*x__2(t)-6*x__3(t), diff(x__2(t),t) = -9*x__1(t)+5*x__2(t)-9*x__3(t), diff(x__3(t),t) = -x__2(t)-x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 146

ode={D[x1[t],t]==-7*x1[t]+6*x2[t]-6*x3[t],D[x2[t],t]==-9*x1[t]+5*x2[t]-9*x3[t],D[x3[t],t]==0*x1[t]-1*x2[t]-1*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t} ((6 c_1-2 c_2+6 c_3) \cos (3 t)-6 (c_1-c_2+c_3) \sin (3 t)-3 c_1+2 c_2-6 c_3) \\ \text {x2}(t)\to e^{-t} (c_2 \cos (3 t)+(-3 c_1+2 c_2-3 c_3) \sin (3 t)) \\ \text {x3}(t)\to \frac {1}{3} e^{-t} ((-3 c_1+2 c_2-3 c_3) \cos (3 t)-c_2 \sin (3 t)+3 c_1-2 c_2+6 c_3) \\ \end{align*}

✓ Sympy. Time used: 0.147 (sec). Leaf size: 90

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(7*x__1(t) - 6*x__2(t) + 6*x__3(t) + Derivative(x__1(t), t),0),Eq(9*x__1(t) - 5*x__2(t) + 9*x__3(t) + Derivative(x__2(t), t),0),Eq(x__2(t) + x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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