Internal problem ID [11020]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin.
CRC Press 2015
Section: Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number: Problem 3(g).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-\frac {x \left (t \right )}{2}+2 y \left (t \right )-3 z \left (t \right )\\ y^{\prime }\left (t \right )&=y \left (t \right )-\frac {z \left (t \right )}{2}\\ z^{\prime }\left (t \right )&=-2 x \left (t \right )+z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 164
dsolve([diff(x(t),t)=-1/2*x(t)+2*y(t)-3*z(t),diff(y(t),t)=y(t)-1/2*z(t),diff(z(t),t)=-2*x(t)+z(t)],[x(t), y(t), z(t)], singsol=all)
\[ x \left (t \right ) = -\frac {c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}+\frac {c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}+\frac {7 c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}}}{8}+\frac {7 c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}}}{8}-c_{1} {\mathrm e}^{3 t} \] \[ y \left (t \right ) = \frac {c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}-\frac {c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}} \sqrt {33}}{8}+\frac {7 c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}}}{8}+\frac {7 c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}}}{8}-\frac {c_{1} {\mathrm e}^{3 t}}{4} \] \[ z \left (t \right ) = c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{\frac {\left (-3+\sqrt {33}\right ) t}{4}}+c_{3} {\mathrm e}^{-\frac {\left (3+\sqrt {33}\right ) t}{4}} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 483
DSolve[{x'[t]==-1/2*x[t]+2*y[t]-3*z[t],y'[t]==y[t]-1/2*z[t],z'[t]==-2*x[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{264} e^{-\frac {1}{4} \left (3+\sqrt {33}\right ) t} \left (c_1 \left (\left (88-16 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}+88 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+88+16 \sqrt {33}\right )+22 (4 c_2-7 c_3) e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+\left (4 \left (3 \sqrt {33}-11\right ) c_2+\left (77-13 \sqrt {33}\right ) c_3\right ) e^{\frac {\sqrt {33} t}{2}}-4 \left (11+3 \sqrt {33}\right ) c_2+\left (77+13 \sqrt {33}\right ) c_3\right ) \\ y(t)\to \frac {e^{-\frac {1}{4} \left (3+\sqrt {33}\right ) t} \left (-4 c_1 \left (\left (11+5 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}-22 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+11-5 \sqrt {33}\right )+22 (4 c_2-7 c_3) e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+\left (\left (484+92 \sqrt {33}\right ) c_2+\left (77+3 \sqrt {33}\right ) c_3\right ) e^{\frac {\sqrt {33} t}{2}}+\left (484-92 \sqrt {33}\right ) c_2+\left (77-3 \sqrt {33}\right ) c_3\right )}{1056} \\ z(t)\to \frac {1}{264} e^{-\frac {1}{4} \left (3+\sqrt {33}\right ) t} \left (c_1 \left (\left (44-12 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}-88 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+44+12 \sqrt {33}\right )-22 (4 c_2-7 c_3) e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+\left (4 \left (11+5 \sqrt {33}\right ) c_2+\left (55-7 \sqrt {33}\right ) c_3\right ) e^{\frac {\sqrt {33} t}{2}}+\left (44-20 \sqrt {33}\right ) c_2+\left (55+7 \sqrt {33}\right ) c_3\right ) \\ \end{align*}