Internal problem ID [13140]
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: 17.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )+3 y\\ y^{\prime }&=-y+z \left (t \right )\\ z^{\prime }\left (t \right )&=5 x \left (t \right )-5 y \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 101
dsolve([diff(x(t),t)=-4*x(t)+3*y(t)+0*z(t),diff(y(t),t)=0*x(t)-1*y(t)+1*z(t),diff(z(t),t)=5*x(t)-5*y(t)+0*z(t)],singsol=all)
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_{1} +\frac {6 c_{2} {\mathrm e}^{-2 t} \sin \left (t \right )}{5}-\frac {3 c_{2} {\mathrm e}^{-2 t} \cos \left (t \right )}{5}+\frac {6 \,{\mathrm e}^{-2 t} \cos \left (t \right ) c_{3}}{5}+\frac {3 \,{\mathrm e}^{-2 t} \sin \left (t \right ) c_{3}}{5} \\ y \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{-2 t} \sin \left (t \right )+{\mathrm e}^{-2 t} \cos \left (t \right ) c_{3} \\ z \left (t \right ) &= -{\mathrm e}^{-2 t} \left (c_{2} \sin \left (t \right )+c_{3} \sin \left (t \right )-c_{2} \cos \left (t \right )+c_{3} \cos \left (t \right )\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 152
DSolve[{x'[t]==-4*x[t]+3*y[t]+0*z[t],y'[t]==0*x[t]-1*y[t]+1*z[t],z'[t]==5*x[t]-5*y[t]+0*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} e^{-2 t} \left ((5 c_1-3 c_2+3 c_3) e^t-3 (c_1-c_2+c_3) \cos (t)-3 (3 c_1-3 c_2+c_3) \sin (t)\right ) \\ y(t)\to \frac {1}{2} e^{-2 t} \left ((5 c_1-3 c_2+3 c_3) e^t+(-5 c_1+5 c_2-3 c_3) \cos (t)-(5 c_1-5 c_2+c_3) \sin (t)\right ) \\ z(t)\to e^{-2 t} (c_3 \cos (t)+(5 c_1-5 c_2+2 c_3) \sin (t)) \\ \end{align*}