Internal problem ID [5987]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.
EXERCISES 8.2. Page 346
Problem number: 14.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )+4 z \left (t \right )\\ y^{\prime }\left (t \right )&=2 y \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )+z \left (t \right ) \end {align*}
With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 3, z \left (0\right ) = 0] \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 53
dsolve([diff(x(t),t) = x(t)+y(t)+4*z(t), diff(y(t),t) = 2*y(t), diff(z(t),t) = x(t)+y(t)+z(t), x(0) = 1, y(0) = 3, z(0) = 0],[x(t), y(t), z(t)], singsol=all)
\[ x \left (t \right ) = -5 \,{\mathrm e}^{2 t}+{\mathrm e}^{-t}+5 \,{\mathrm e}^{3 t} \] \[ y \left (t \right ) = 3 \,{\mathrm e}^{2 t} \] \[ z \left (t \right ) = -2 \,{\mathrm e}^{2 t}-\frac {{\mathrm e}^{-t}}{2}+\frac {5 \,{\mathrm e}^{3 t}}{2} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 62
DSolve[{x'[t]==x[t]+y[t]+4*z[t],y'[t]==2*y[t],z'[t]==x[t]+y[t]+z[t]},{x[0]==1,y[0]==3,z[0]==0},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{-t}-5 e^{2 t}+5 e^{3 t} \\ y(t)\to 3 e^{2 t} \\ z(t)\to \frac {1}{2} e^{-t} \left (e^{3 t} \left (5 e^t-4\right )-1\right ) \\ \end{align*}