Internal problem ID [13132]
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: 7.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )\\ y^{\prime }&=2 y-z \left (t \right )\\ z^{\prime }\left (t \right )&=-y+2 z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 38
dsolve([diff(x(t),t)=1*x(t)+0*y(t)+0*z(t),diff(y(t),t)=0*x(t)+2*y(t)-1*z(t),diff(z(t),t)=0*x(t)-1*y(t)+2*z(t)],singsol=all)
\begin{align*} x \left (t \right ) &= c_{3} {\mathrm e}^{t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{3 t} \\ z \left (t \right ) &= c_{1} {\mathrm e}^{t}-c_{2} {\mathrm e}^{3 t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 144
DSolve[{x'[t]==1*x[t]+0*y[t]+0*z[t],y'[t]==0*x[t]+2*y[t]-1*z[t],z'[t]==0*x[t]-1*y[t]+2*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to c_1 e^t \\ y(t)\to \frac {1}{2} e^t \left (c_2 e^{2 t}-c_3 e^{2 t}+c_2+c_3\right ) \\ z(t)\to \frac {1}{2} e^t \left (c_2 \left (-e^{2 t}\right )+c_3 e^{2 t}+c_2+c_3\right ) \\ x(t)\to 0 \\ y(t)\to \frac {1}{2} e^t \left (c_2 e^{2 t}-c_3 e^{2 t}+c_2+c_3\right ) \\ z(t)\to \frac {1}{2} e^t \left (c_2 \left (-e^{2 t}\right )+c_3 e^{2 t}+c_2+c_3\right ) \\ \end{align*}