7.23.12 problem 18

Internal problem ID [598]
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 5.2 (Applications). Problems at page 345
Problem number : 18
Date solved : Monday, January 27, 2025 at 02:55:09 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 60

dsolve([diff(x(t),t)=x(t)+2*y(t)+z(t),diff(y(t),t)=6*x(t)-y(t),diff(z(t),t)=-x(t)-2*y(t)-z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{-4 t}}{2}+\frac {2 c_3 \,{\mathrm e}^{3 t}}{3}+\frac {c_1}{6} \\ y \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{-4 t}+c_3 \,{\mathrm e}^{3 t} \\ z \left (t \right ) &= \frac {c_2 \,{\mathrm e}^{-4 t}}{2}-\frac {2 c_3 \,{\mathrm e}^{3 t}}{3}-\frac {13 c_1}{6} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 190

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

\begin{align*} x(t)\to \frac {1}{84} e^{-4 t} \left (c_1 \left (-7 e^{4 t}+64 e^{7 t}+27\right )+24 c_2 \left (e^{7 t}-1\right )+c_3 \left (-7 e^{4 t}+16 e^{7 t}-9\right )\right ) \\ y(t)\to \frac {1}{14} e^{-4 t} \left (c_1 \left (-7 e^{4 t}+16 e^{7 t}-9\right )+c_2 \left (6 e^{7 t}+8\right )+c_3 \left (-7 e^{4 t}+4 e^{7 t}+3\right )\right ) \\ z(t)\to \frac {1}{84} e^{-4 t} \left (c_1 \left (91 e^{4 t}-64 e^{7 t}-27\right )-24 c_2 \left (e^{7 t}-1\right )-c_3 \left (-91 e^{4 t}+16 e^{7 t}-9\right )\right ) \\ \end{align*}