14.29.8 problem 8

Internal problem ID [2806]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 8
Date solved : Monday, January 27, 2025 at 06:14:03 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+y+z \left (t \right )\\ y^{\prime }&=-3 x \left (t \right )+2 y+3 z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )-y-2 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 53

dsolve([diff(x(t),t)=-2*x(t)+1*y(t)+1*z(t),diff(y(t),t)=-3*x(t)+2*y(t)+3*z(t),diff(z(t),t)=1*x(t)-1*y(t)-2*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_2 +c_3 \,{\mathrm e}^{-t} \\ y &= 3 c_2 +3 c_3 \,{\mathrm e}^{-t}+{\mathrm e}^{-t} c_1 \\ z \left (t \right ) &= -2 c_3 \,{\mathrm e}^{-t}-c_2 -{\mathrm e}^{-t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 106

DSolve[{D[x[t],t]==-2*x[t]+1*y[t]+1*z[t],D[y[t],t]==-3*x[t]+2*y[t]+3*z[t],D[z[t],t]==1*x[t]-1*y[t]-2*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-t} \left ((c_2+c_3) \left (e^t-1\right )-c_1 \left (e^t-2\right )\right ) \\ y(t)\to e^{-t} \left (-3 c_1 \left (e^t-1\right )+c_2 \left (3 e^t-2\right )+3 c_3 \left (e^t-1\right )\right ) \\ z(t)\to e^{-t} \left (c_1 \left (e^t-1\right )-c_2 \left (e^t-1\right )-c_3 \left (e^t-2\right )\right ) \\ \end{align*}