Internal problem ID [5983]
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: 11.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )-y \left (t \right )\\ y^{\prime }\left (t \right )&=\frac {3 x \left (t \right )}{4}-\frac {3 y \left (t \right )}{2}+3 z \left (t \right )\\ z^{\prime }\left (t \right )&=\frac {x \left (t \right )}{8}+\frac {y \left (t \right )}{4}-\frac {z \left (t \right )}{2} \end {align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 67
dsolve([diff(x(t),t)=-x(t)-y(t),diff(y(t),t)=3/4*x(t)-3/2*y(t)+3*z(t),diff(z(t),t)=1/8*x(t)+1/4*y(t)-1/2*z(t)],[x(t), y(t), z(t)], singsol=all)
\[ x \left (t \right ) = -\frac {12 c_{1} {\mathrm e}^{-\frac {t}{2}}}{5}-4 \,{\mathrm e}^{-t} c_{2} -4 c_{3} {\mathrm e}^{-\frac {3 t}{2}} \] \[ y \left (t \right ) = \frac {6 c_{1} {\mathrm e}^{-\frac {t}{2}}}{5}-2 c_{3} {\mathrm e}^{-\frac {3 t}{2}} \] \[ z \left (t \right ) = c_{1} {\mathrm e}^{-\frac {t}{2}}+{\mathrm e}^{-t} c_{2} +c_{3} {\mathrm e}^{-\frac {3 t}{2}} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 162
DSolve[{x'[t]==-x[t]-y[t],y'[t]==3/4*x[t]-3/2*y[t]+3*z[t],z'[t]==1/8x[t]+1/4*y[t]-1/2*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} e^{-3 t/2} \left (c_1 \left (8 e^{t/2}-3 e^t-3\right )-4 \left (e^{t/2}-1\right ) \left (3 c_3 \left (e^{t/2}-1\right )+c_2\right )\right ) \\ y(t)\to \frac {1}{4} e^{-3 t/2} \left (3 c_1 \left (e^t-1\right )+4 \left (3 c_3 \left (e^t-1\right )+c_2\right )\right ) \\ z(t)\to \frac {1}{8} e^{-3 t/2} \left (5 (c_1+4 c_3) e^t-4 (2 c_1-c_2+6 c_3) e^{t/2}+3 c_1-4 c_2+12 c_3\right ) \\ \end{align*}