7.24.19 problem 29 and 38

Internal problem ID [619]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.3 (Matrices and linear systems). Problems at page 364
Problem number : 29 and 38
Date solved : Saturday, March 29, 2025 at 05:00:52 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-8 x_{1} \left (t \right )-11 x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=6 x_{1} \left (t \right )+9 x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-6 x_{1} \left (t \right )-6 x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 5\\ x_{2} \left (0\right ) = -7\\ x_{3} \left (0\right ) = 11 \end{align*}

✓ Maple. Time used: 0.136 (sec). Leaf size: 55

ode:=[diff(x__1(t),t) = -8*x__1(t)-11*x__2(t)-2*x__3(t), diff(x__2(t),t) = 6*x__1(t)+9*x__2(t)+2*x__3(t), diff(x__3(t),t) = -6*x__1(t)-6*x__2(t)+x__3(t)]; 
ic:=x__1(0) = 5x__2(0) = -7x__3(0) = 11; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= 15 \,{\mathrm e}^{t}-6 \,{\mathrm e}^{-2 t}-4 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= -15 \,{\mathrm e}^{t}+4 \,{\mathrm e}^{-2 t}+4 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= 15 \,{\mathrm e}^{t}-4 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 64

ode={D[x1[t],t]==-8*x1[t]-11*x2[t]-2*x3[t],D[x2[t],t]==6*x1[t]+9*x2[t]+2*x3[t],D[x3[t],t]==-6*x1[t]-6*x2[t]+x3[t]}; 
ic={x1[0]==5,x2[0]==-7,x3[0]==11}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -6 e^{-2 t}+15 e^t-4 e^{3 t} \\ \text {x2}(t)\to 4 e^{-2 t}-15 e^t+4 e^{3 t} \\ \text {x3}(t)\to 15 e^t-4 e^{-2 t} \\ \end{align*}

✓ Sympy. Time used: 0.154 (sec). Leaf size: 58

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) + 11*x__2(t) + 2*x__3(t) + Derivative(x__1(t), t),0),Eq(-6*x__1(t) - 9*x__2(t) - 2*x__3(t) + Derivative(x__2(t), t),0),Eq(6*x__1(t) + 6*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)
 

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