Internal problem ID [13128]
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: 1.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=\frac {y}{10}\\ y^{\prime }&=\frac {z \left (t \right )}{5}\\ z^{\prime }\left (t \right )&=\frac {2 x \left (t \right )}{5} \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 183
dsolve([diff(x(t),t)=0*x(t)+1/10*y(t)+0*z(t),diff(y(t),t)=0*x(t)+0*y(t)+2/10*z(t),diff(z(t),t)=4/10*x(t)+0*y(t)+0*z(t)],singsol=all)
\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{\frac {t}{5}} c_{1}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{4}+\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{4}-\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{4}-\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{4} \\ y \left (t \right ) &= {\mathrm e}^{\frac {t}{5}} c_{1} -\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{2}-\frac {c_{2} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{2}-\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{2}+\frac {c_{3} {\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{2} \\ z \left (t \right ) &= {\mathrm e}^{\frac {t}{5}} c_{1} +c_{2} {\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )+c_{3} {\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.059 (sec). Leaf size: 269
DSolve[{x'[t]==0*x[t]+1/10*y[t]+0*z[t],y'[t]==0*x[t]+0*y[t]+2/10*z[t],z'[t]==4/10*x[t]+0*y[t]+0*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{6} e^{-t/10} \left ((2 c_1+c_2+c_3) e^{t/10} \sqrt [5]{e^t}+(4 c_1-c_2-c_3) \cos \left (\frac {\sqrt {3} t}{10}\right )+\sqrt {3} (c_2-c_3) \sin \left (\frac {\sqrt {3} t}{10}\right )\right ) \\ y(t)\to \frac {1}{3} e^{-t/10} \left ((2 c_1+c_2+c_3) e^{t/10} \sqrt [5]{e^t}-(2 c_1-2 c_2+c_3) \cos \left (\frac {\sqrt {3} t}{10}\right )-\sqrt {3} (2 c_1-c_3) \sin \left (\frac {\sqrt {3} t}{10}\right )\right ) \\ z(t)\to \frac {1}{3} e^{-t/10} \left ((2 c_1+c_2+c_3) e^{t/10} \sqrt [5]{e^t}-(2 c_1+c_2-2 c_3) \cos \left (\frac {\sqrt {3} t}{10}\right )+\sqrt {3} (2 c_1-c_2) \sin \left (\frac {\sqrt {3} t}{10}\right )\right ) \\ \end{align*}