14.29.5 problem 5

Internal problem ID [2803]
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 : 5
Date solved : Monday, January 27, 2025 at 06:14:01 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-7 x \left (t \right )+y-6 z \left (t \right )\\ y^{\prime }&=10 x \left (t \right )-4 y+12 z \left (t \right )\\ z^{\prime }\left (t \right )&=2 x \left (t \right )-y+z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 73

dsolve([diff(x(t),t)=-7*x(t)+1*y(t)-6*z(t),diff(y(t),t)=10*x(t)-4*y(t)+12*z(t),diff(z(t),t)=2*x(t)-1*y(t)+1*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-5 t}+{\mathrm e}^{-2 t} c_2 +c_3 \,{\mathrm e}^{-3 t} \\ y &= -2 c_1 \,{\mathrm e}^{-5 t}-{\mathrm e}^{-2 t} c_2 -2 c_3 \,{\mathrm e}^{-3 t} \\ z \left (t \right ) &= -\frac {2 c_1 \,{\mathrm e}^{-5 t}}{3}-{\mathrm e}^{-2 t} c_2 -c_3 \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 166

DSolve[{D[x[t],t]==-7*x[t]+1*y[t]-6*z[t],D[y[t],t]==10*x[t]-4*y[t]+12*z[t],D[z[t],t]==2*x[t]-1*y[t]+1*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-5 t} \left (c_1 \left (-4 e^{2 t}+2 e^{3 t}+3\right )+\left (e^t-1\right ) \left (c_2 e^{2 t}-3 c_3 e^t-3 c_3\right )\right ) \\ y(t)\to e^{-5 t} \left (-2 c_1 \left (-4 e^{2 t}+e^{3 t}+3\right )-c_2 e^{3 t}+2 (c_2+3 c_3) e^{2 t}-6 c_3\right ) \\ z(t)\to e^{-5 t} \left (-2 c_1 \left (-2 e^{2 t}+e^{3 t}+1\right )-c_2 e^{3 t}+(c_2+3 c_3) e^{2 t}-2 c_3\right ) \\ \end{align*}