Internal problem ID [13139]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 3. Linear Systems. Exercises section 3.8 page 371
Problem number: 16.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-y\\ y^{\prime }&=-2 y+3 z \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+3 y-z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 171
dsolve([diff(x(t),t)=2*x(t)-1*y(t)+0*z(t),diff(y(t),t)=0*x(t)-2*y(t)+3*z(t),diff(z(t),t)=-1*x(t)+3*y(t)-1*z(t)],singsol=all)
\begin{align*} x \left (t \right ) &= -c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t}-c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t}-\frac {2 c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+\frac {2 c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+c_{1} {\mathrm e}^{t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t}+c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t} \\ z \left (t \right ) &= \frac {2 c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}-\frac {2 c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t} \sqrt {3}}{3}+\frac {c_{2} {\mathrm e}^{\left (-1+2 \sqrt {3}\right ) t}}{3}+\frac {c_{3} {\mathrm e}^{-\left (1+2 \sqrt {3}\right ) t}}{3}+c_{1} {\mathrm e}^{t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 474
DSolve[{x'[t]==2*x[t]-1*y[t]+0*z[t],y'[t]==0*x[t]-2*y[t]+3*z[t],z'[t]==-1*x[t]+3*y[t]-1*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{16} e^{-\left (\left (1+2 \sqrt {3}\right ) t\right )} \left (c_1 \left (\left (5+3 \sqrt {3}\right ) e^{4 \sqrt {3} t}+6 e^{2 \left (1+\sqrt {3}\right ) t}+5-3 \sqrt {3}\right )-2 c_2 \left (\left (1+\sqrt {3}\right ) e^{4 \sqrt {3} t}-2 e^{2 \left (1+\sqrt {3}\right ) t}+1-\sqrt {3}\right )-c_3 \left (\left (3+\sqrt {3}\right ) e^{4 \sqrt {3} t}-6 e^{2 \left (1+\sqrt {3}\right ) t}+3-\sqrt {3}\right )\right ) \\ y(t)\to \frac {1}{16} e^{-\left (\left (1+2 \sqrt {3}\right ) t\right )} \left (c_1 \left (-\left (3+\sqrt {3}\right ) e^{4 \sqrt {3} t}+6 e^{2 \left (1+\sqrt {3}\right ) t}-3+\sqrt {3}\right )+2 c_2 \left (-\left (\sqrt {3}-3\right ) e^{4 \sqrt {3} t}+2 e^{2 \left (1+\sqrt {3}\right ) t}+3+\sqrt {3}\right )+3 c_3 \left (\left (\sqrt {3}-1\right ) e^{4 \sqrt {3} t}+2 e^{2 \left (1+\sqrt {3}\right ) t}-1-\sqrt {3}\right )\right ) \\ z(t)\to -\frac {1}{48} e^{-\left (\left (1+2 \sqrt {3}\right ) t\right )} \left (c_1 \left (\left (9+7 \sqrt {3}\right ) e^{4 \sqrt {3} t}-18 e^{2 \left (1+\sqrt {3}\right ) t}+9-7 \sqrt {3}\right )-2 c_2 \left (\left (5 \sqrt {3}-3\right ) e^{4 \sqrt {3} t}+6 e^{2 \left (1+\sqrt {3}\right ) t}-3-5 \sqrt {3}\right )+3 c_3 \left (\left (\sqrt {3}-5\right ) e^{4 \sqrt {3} t}-6 e^{2 \left (1+\sqrt {3}\right ) t}-5-\sqrt {3}\right )\right ) \\ \end{align*}