Internal problem ID [12368]
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-3 z \left (t \right )\\ y^{\prime }&=y-\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: 0.031 (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)],singsol=all)
\begin{align*} 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}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.053 (sec). Leaf size: 523
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 )+4 c_2 \left (\left (3 \sqrt {33}-11\right ) e^{\frac {\sqrt {33} t}{2}}+22 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}-11-3 \sqrt {33}\right )-c_3 \left (\left (13 \sqrt {33}-77\right ) e^{\frac {\sqrt {33} t}{2}}+154 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}-77-13 \sqrt {33}\right )\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 )+c_2 \left (\left (484+92 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}+88 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+484-92 \sqrt {33}\right )+c_3 \left (\left (77+3 \sqrt {33}\right ) e^{\frac {\sqrt {33} t}{2}}-154 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}+77-3 \sqrt {33}\right )\right )}{1056} \\ z(t)\to -\frac {1}{264} e^{-\frac {1}{4} \left (3+\sqrt {33}\right ) t} \left (4 c_1 \left (\left (3 \sqrt {33}-11\right ) e^{\frac {\sqrt {33} t}{2}}+22 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}-11-3 \sqrt {33}\right )-4 c_2 \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 )+c_3 \left (\left (7 \sqrt {33}-55\right ) e^{\frac {\sqrt {33} t}{2}}-154 e^{\frac {1}{4} \left (15+\sqrt {33}\right ) t}-55-7 \sqrt {33}\right )\right ) \\ \end{align*}