20.19.8 problem 9

Internal problem ID [3888]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.8 (Matrix exponential function), page 642
Problem number : 9
Date solved : Sunday, March 30, 2025 at 02:11:06 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-8 x_{1} \left (t \right )+6 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-12 x_{1} \left (t \right )+10 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.194 (sec). Leaf size: 73

ode:=[diff(x__1(t),t) = -8*x__1(t)+6*x__2(t)-3*x__3(t), diff(x__2(t),t) = -12*x__1(t)+10*x__2(t)-3*x__3(t), diff(x__3(t),t) = -2*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-2 t} c_2 +{\mathrm e}^{4 t} c_1 -\frac {c_3 \,{\mathrm e}^{-2 t}}{2}-3 \,{\mathrm e}^{-2 t} c_3 t \\ x_{2} \left (t \right ) &= {\mathrm e}^{-2 t} c_2 +2 \,{\mathrm e}^{4 t} c_1 -\frac {c_3 \,{\mathrm e}^{-2 t}}{2}-3 \,{\mathrm e}^{-2 t} c_3 t \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 105

ode={D[x1[t],t]==-8*x1[t]+6*x2[t]-3*x3[t],D[x2[t],t]==-12*x1[t]+10*x2[t]-3*x3[t],D[x3[t],t]==-12*x1[t]+12*x2[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} (-6 c_1 t+6 c_2 t-3 c_3 t+c_1) \\ \text {x2}(t)\to e^{-2 t} \left (-\left (c_1 \left (6 t+e^{6 t}-1\right )\right )+c_2 e^{6 t}+6 c_2 t-3 c_3 t\right ) \\ \text {x3}(t)\to e^{-2 t} \left (-2 c_1 \left (e^{6 t}-1\right )+2 c_2 \left (e^{6 t}-1\right )+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.136 (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(8*x__1(t) - 6*x__2(t) + 3*x__3(t) + Derivative(x__1(t), t),0),Eq(12*x__1(t) - 10*x__2(t) + 3*x__3(t) + Derivative(x__2(t), t),0),Eq(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)
 

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