Internal problem ID [12811]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 3. Linear Systems. Exercises section 3.8 page 371
Problem number: 6.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+3 z \left (t \right )\\ y^{\prime }&=-y\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )+z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 51
dsolve([diff(x(t),t)=1*x(t)+0*y(t)+3*z(t),diff(y(t),t)=0*x(t)-1*y(t)+0*z(t),diff(z(t),t)=-3*x(t)+0*y(t)+1*z(t)],[x(t), y(t), z(t)], singsol=all)
\begin{align*} x \left (t \right ) = -{\mathrm e}^{t} \left (c_{2} \cos \left (3 t \right )-\sin \left (3 t \right ) c_{3} \right ) y \left (t \right ) = c_{1} {\mathrm e}^{-t} z \left (t \right ) = {\mathrm e}^{t} \left (c_{3} \cos \left (3 t \right )+\sin \left (3 t \right ) c_{2} \right ) \end{align*}
✓ Solution by Mathematica
Time used: 0.032 (sec). Leaf size: 108
DSolve[{x'[t]==1*x[t]+0*y[t]+3*z[t],y'[t]==0*x[t]-1*y[t]+0*z[t],z'[t]==-3*x[t]+0*y[t]+1*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^t (c_1 \cos (3 t)+c_2 \sin (3 t)) z(t)\to e^t (c_2 \cos (3 t)-c_1 \sin (3 t)) y(t)\to c_3 e^{-t} x(t)\to e^t (c_1 \cos (3 t)+c_2 \sin (3 t)) z(t)\to e^t (c_2 \cos (3 t)-c_1 \sin (3 t)) y(t)\to 0 \end{align*}