20.16.12 problem 12

Internal problem ID [3845]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 12
Date solved : Tuesday, March 04, 2025 at 05:17:57 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.135 (sec). Leaf size: 91

ode:=[diff(x__1(t),t) = 3*x__1(t)-x__3(t), diff(x__2(t),t) = -3*x__2(t)-x__3(t), diff(x__3(t),t) = 2*x__2(t)-x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {5 \,{\mathrm e}^{-2 t} \sin \left (t \right ) c_{2}}{26}-\frac {{\mathrm e}^{-2 t} \sin \left (t \right ) c_3}{26}+\frac {c_{2} {\mathrm e}^{-2 t} \cos \left (t \right )}{26}+\frac {5 c_3 \,{\mathrm e}^{-2 t} \cos \left (t \right )}{26}+c_{1} {\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \left (\sin \left (t \right ) c_{2} +c_3 \sin \left (t \right )-c_{2} \cos \left (t \right )+\cos \left (t \right ) c_3 \right )}{2} \\ x_{3} \left (t \right ) &= {\mathrm e}^{-2 t} \left (\sin \left (t \right ) c_{2} +\cos \left (t \right ) c_3 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 105

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

\begin{align*} \text {x1}(t)\to \frac {1}{13} e^{-2 t} \left ((13 c_1-c_2-3 c_3) e^{5 t}+(c_2+3 c_3) \cos (t)+(5 c_2+2 c_3) \sin (t)\right ) \\ \text {x2}(t)\to e^{-2 t} (c_2 \cos (t)-(c_2+c_3) \sin (t)) \\ \text {x3}(t)\to e^{-2 t} (c_3 \cos (t)+(2 c_2+c_3) \sin (t)) \\ \end{align*}

✓ Sympy. Time used: 0.180 (sec). Leaf size: 97

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*x__1(t) + x__3(t) + Derivative(x__1(t), t),0),Eq(3*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(-2*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)
 

\[ \left [ x^{1}{\left (t \right )} = C_{3} e^{3 t} - \left (\frac {C_{1}}{26} - \frac {5 C_{2}}{26}\right ) e^{- 2 t} \cos {\left (t \right )} - \left (\frac {5 C_{1}}{26} + \frac {C_{2}}{26}\right ) e^{- 2 t} \sin {\left (t \right )}, \ x^{2}{\left (t \right )} = \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- 2 t} \sin {\left (t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- 2 t} \cos {\left (t \right )}, \ x^{3}{\left (t \right )} = - C_{1} e^{- 2 t} \sin {\left (t \right )} + C_{2} e^{- 2 t} \cos {\left (t \right )}\right ] \]