Internal problem ID [11967]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 7, Systems of linear differential equations. Section 7.7. Exercises page
375
Problem number: 2.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )-z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+3 y \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )-6 y \left (t \right )+6 z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 74
dsolve([diff(x(t),t)=x(t)-y(t)-z(t),diff(y(t),t)=x(t)+3*y(t)+z(t),diff(z(t),t)=-3*x(t)-6*y(t)+6*z(t)],singsol=all)
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{5 t} \\ y \left (t \right ) &= -c_{1} {\mathrm e}^{3 t}-\frac {7 c_{2} {\mathrm e}^{2 t}}{10}-c_{3} {\mathrm e}^{5 t} \\ z \left (t \right ) &= -c_{1} {\mathrm e}^{3 t}-\frac {3 c_{2} {\mathrm e}^{2 t}}{10}-3 c_{3} {\mathrm e}^{5 t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 217
DSolve[{x'[t]==x[t]-y[t]-z[t],y'[t]==x[t]+3*y[t]+z[t],z'[t]==3*x[t]-6*y[t]+6*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {1}{45} e^{2 t} \left (5 (c_1+10 c_2) e^{2 t} \cos \left (\sqrt {5} t\right )+\sqrt {5} (7 c_1-11 c_2+9 c_3) e^{2 t} \sin \left (\sqrt {5} t\right )-50 (c_1+c_2)\right ) \\ y(t)\to \frac {1}{45} e^{2 t} \left (5 (c_1+10 c_2) e^{2 t} \cos \left (\sqrt {5} t\right )+\sqrt {5} (7 c_1-11 c_2+9 c_3) e^{2 t} \sin \left (\sqrt {5} t\right )-5 (c_1+c_2)\right ) \\ z(t)\to (c_1+c_2) \left (-e^{2 t}\right )+(c_1+c_2+c_3) e^{4 t} \cos \left (\sqrt {5} t\right )+\frac {(c_1-8 c_2+2 c_3) e^{4 t} \sin \left (\sqrt {5} t\right )}{\sqrt {5}} \\ \end{align*}