20.16.9 problem 9

Internal problem ID [3842]
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 : 9
Date solved : Tuesday, March 04, 2025 at 05:17:53 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 116

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

\begin{align*} \text {x1}(t)\to e^{2 t} (c_1 \cos (3 t)+c_2 \sin (3 t)) \\ \text {x3}(t)\to e^{2 t} (c_2 \cos (3 t)-c_1 \sin (3 t)) \\ \text {x2}(t)\to c_3 e^{-4 t} \\ \text {x1}(t)\to e^{2 t} (c_1 \cos (3 t)+c_2 \sin (3 t)) \\ \text {x3}(t)\to e^{2 t} (c_2 \cos (3 t)-c_1 \sin (3 t)) \\ \text {x2}(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.138 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-2*x__1(t) - 3*x__3(t) + Derivative(x__1(t), t),0),Eq(4*x__2(t) + Derivative(x__2(t), t),0),Eq(3*x__1(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)
 

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