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 : Monday, January 27, 2025 at 02:56:25 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 55

dsolve([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), x__1(0) = 5, x__2(0) = -7, x__3(0) = 11], singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{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]},{x1[0]==5,x2[0]==-7,x3[0]==11},{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*}