Internal problem ID [5973]
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 8.1. Page 332
Problem number: 16.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=x \relax (t )+z \relax (t )\\ y^{\prime }\relax (t )&=x \relax (t )+y \relax (t )\\ z^{\prime }\relax (t )&=-2 x \relax (t )-z \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.176 (sec). Leaf size: 56
dsolve([diff(x(t),t)=x(t)+z(t),diff(y(t),t)=x(t)+y(t),diff(z(t),t)=-2*x(t)-z(t)],[x(t), y(t), z(t)], singsol=all)
\[ x \relax (t ) = -\frac {c_{2} \cos \relax (t )}{2}+\frac {c_{3} \sin \relax (t )}{2}-\frac {\sin \relax (t ) c_{2}}{2}-\frac {c_{3} \cos \relax (t )}{2} \] \[ y \relax (t ) = \frac {c_{2} \cos \relax (t )}{2}-\frac {c_{3} \sin \relax (t )}{2}+c_{1} {\mathrm e}^{t} \] \[ z \relax (t ) = \sin \relax (t ) c_{2}+c_{3} \cos \relax (t ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 76
DSolve[{x'[t]==x[t]+z[t],y'[t]==x[t]+y[t],z'[t]==-2*x[t]-z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to c_1 \cos (t)+(c_1+c_3) \sin (t) \\ y(t)\to c_2 e^t+c_1 \left (e^t-\cos (t)\right )-\frac {1}{2} c_3 \left (-e^t+\sin (t)+\cos (t)\right ) \\ z(t)\to c_3 \cos (t)-(2 c_1+c_3) \sin (t) \\ \end{align*}