Internal problem ID [13131]
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: 0.016 (sec). Leaf size: 50
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)],singsol=all)
\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right )\right ) \\ y \left (t \right ) &= c_{3} {\mathrm e}^{-t} \\ z \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \cos \left (3 t \right )-c_{2} \sin \left (3 t \right )\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*}