Internal problem ID [12815]
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: 12.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+2 y\\ y^{\prime }&=2 x \left (t \right )-4 y\\ z^{\prime }\left (t \right )&=-z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 36
dsolve([diff(x(t),t)=-1*x(t)+2*y(t)+0*z(t),diff(y(t),t)=2*x(t)-4*y(t)+0*z(t),diff(z(t),t)=0*x(t)+0*y(t)-1*z(t)],[x(t), y(t), z(t)], singsol=all)
\begin{align*} x \left (t \right ) = -\frac {c_{2} {\mathrm e}^{-5 t}}{2}+2 c_{1} y \left (t \right ) = c_{1} +c_{2} {\mathrm e}^{-5 t} z \left (t \right ) = c_{3} {\mathrm e}^{-t} \end{align*}
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 158
DSolve[{x'[t]==-1*x[t]+2*y[t]+0*z[t],y'[t]==2*x[t]-4*y[t]+0*z[t],z'[t]==0*x[t]+0*y[t]-1*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{5} e^{-5 t} \left (c_1 \left (4 e^{5 t}+1\right )+2 c_2 \left (e^{5 t}-1\right )\right ) y(t)\to \frac {1}{5} e^{-5 t} \left (2 c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}+4\right )\right ) z(t)\to c_3 e^{-t} x(t)\to \frac {1}{5} e^{-5 t} \left (c_1 \left (4 e^{5 t}+1\right )+2 c_2 \left (e^{5 t}-1\right )\right ) y(t)\to \frac {1}{5} e^{-5 t} \left (2 c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}+4\right )\right ) z(t)\to 0 \end{align*}