Internal problem ID [5784]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page
400
Problem number: 3(e).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=3 x \relax (t )+2 y \relax (t )+z \relax (t )\\ y^{\prime }\relax (t )&=-2 x \relax (t )-y \relax (t )+3 z \relax (t )\\ z^{\prime }\relax (t )&=x \relax (t )+y \relax (t )+z \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 63
dsolve([diff(x(t),t)=3*x(t)+2*y(t)+z(t),diff(y(t),t)=-2*x(t)-y(t)+3*z(t),diff(z(t),t)=x(t)+y(t)+z(t)],[x(t), y(t), z(t)], singsol=all)
\[ x \relax (t ) = \frac {5 c_{3} {\mathrm e}^{3 t}}{2}-c_{1} {\mathrm e}^{t}+\frac {3 \,{\mathrm e}^{-t} c_{2}}{2} \] \[ y \relax (t ) = -\frac {c_{3} {\mathrm e}^{3 t}}{2}+c_{1} {\mathrm e}^{t}-\frac {7 \,{\mathrm e}^{-t} c_{2}}{2} \] \[ z \relax (t ) = {\mathrm e}^{-t} c_{2}+c_{3} {\mathrm e}^{3 t} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 143
DSolve[{x'[t]==3*x[t]+2*y[t]+z[t],y'[t]==-2*x[t]-y[t]+3*z[t],z'[t]==x[t]+y[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{4} e^t (2 \sinh (t) (2 (2 c_2+c_3) \cosh (t)+(c_2+8 c_3) \sinh (t))+c_1 (4 \sinh (2 t)+\cosh (2 t)+3)) \\ y(t)\to \frac {1}{8} e^{-t} \left (-2 (3 c_1-c_2-8 c_3) e^{2 t}-(c_1+c_2+2 c_3) e^{4 t}+7 (c_1+c_2-2 c_3)\right ) \\ z(t)\to \frac {1}{2} e^t (2 c_3 \cosh (2 t)+(c_1+c_2) \sinh (2 t)) \\ \end{align*}