76.26.6 problem 6

Internal problem ID [17767]
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 : 6
Date solved : Monday, March 31, 2025 at 04:27:06 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=\frac {4 x_{1} \left (t \right )}{3}+\frac {4 x_{2} \left (t \right )}{3}-\frac {11 x_{3} \left (t \right )}{3}\\ \frac {d}{d t}x_{2} \left (t \right )&=-\frac {16 x_{1} \left (t \right )}{3}-\frac {x_{2} \left (t \right )}{3}+\frac {14 x_{3} \left (t \right )}{3}\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )-2 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.143 (sec). Leaf size: 83

ode:=[diff(x__1(t),t) = 4/3*x__1(t)+4/3*x__2(t)-11/3*x__3(t), diff(x__2(t),t) = -16/3*x__1(t)-1/3*x__2(t)+14/3*x__3(t), diff(x__3(t),t) = 3*x__1(t)-2*x__2(t)-2*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \sin \left (5 t \right )+c_3 \cos \left (5 t \right ) \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}-c_2 \sin \left (5 t \right )-c_3 \cos \left (5 t \right )+c_2 \cos \left (5 t \right )-c_3 \sin \left (5 t \right ) \\ x_{3} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}-c_2 \cos \left (5 t \right )+c_3 \sin \left (5 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 170

ode={D[x1[t],t]==4/3*x1[t]+4/3*x2[t]-11/3*x3[t],D[x2[t],t]==-16/3*x1[t]-1/3*x2[t]+14/3*x3[t],D[x3[t],t]==3*x1[t]-2*x2[t]-2*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} \left ((2 c_1-c_2-c_3) e^t \cos (5 t)+(c_1+c_2-2 c_3) e^t \sin (5 t)+c_1+c_2+c_3\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-t} \left (-(c_1-2 c_2+c_3) e^t \cos (5 t)-3 (c_1-c_3) e^t \sin (5 t)+c_1+c_2+c_3\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{-t} \left (-(c_1+c_2-2 c_3) e^t \cos (5 t)+(2 c_1-c_2-c_3) e^t \sin (5 t)+c_1+c_2+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.156 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-4*x__1(t)/3 - 4*x__2(t)/3 + 11*x__3(t)/3 + Derivative(x__1(t), t),0),Eq(16*x__1(t)/3 + x__2(t)/3 - 14*x__3(t)/3 + Derivative(x__2(t), t),0),Eq(-3*x__1(t) + 2*x__2(t) + 2*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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