14.29.6 problem 6

Internal problem ID [2804]
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 : 6
Date solved : Monday, January 27, 2025 at 06:14:01 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=3 x \left (t \right )+2 y+4 z \left (t \right )\\ y^{\prime }&=2 x \left (t \right )+2 z \left (t \right )\\ z^{\prime }\left (t \right )&=4 x \left (t \right )+2 y+3 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 66

dsolve([diff(x(t),t)=3*x(t)+2*y(t)+4*z(t),diff(y(t),t)=2*x(t)-0*y(t)+2*z(t),diff(z(t),t)=4*x(t)+2*y(t)+3*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= 2 c_2 \,{\mathrm e}^{8 t}+2 c_3 \,{\mathrm e}^{-t}+{\mathrm e}^{-t} c_1 \\ y &= c_2 \,{\mathrm e}^{8 t}+c_3 \,{\mathrm e}^{-t} \\ z \left (t \right ) &= 2 c_2 \,{\mathrm e}^{8 t}-\frac {5 c_3 \,{\mathrm e}^{-t}}{2}-{\mathrm e}^{-t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 135

DSolve[{D[x[t],t]==3*x[t]+2*y[t]+4*z[t],D[y[t],t]==2*x[t]-0*y[t]+2*z[t],D[z[t],t]==4*x[t]+2*y[t]+3*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (4 e^{9 t}+5\right )+2 (c_2+2 c_3) \left (e^{9 t}-1\right )\right ) \\ y(t)\to \frac {1}{9} e^{-t} \left (2 c_1 \left (e^{9 t}-1\right )+c_2 \left (e^{9 t}+8\right )+2 c_3 \left (e^{9 t}-1\right )\right ) \\ z(t)\to \frac {1}{9} e^{-t} \left (4 c_1 \left (e^{9 t}-1\right )+2 c_2 \left (e^{9 t}-1\right )+c_3 \left (4 e^{9 t}+5\right )\right ) \\ \end{align*}