Internal problem ID [13107]
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 3.4 page 310
Problem number: 12.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=2 y\\ y^{\prime }&=-2 x \left (t \right )-y \end {align*}
With initial conditions \[ [x \left (0\right ) = -1, y \left (0\right ) = 1] \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 63
dsolve([diff(x(t),t) = 2*y(t), diff(y(t),t) = -2*x(t)-y(t), x(0) = -1, y(0) = 1], singsol=all)
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (\frac {\sqrt {15}\, \sin \left (\frac {t \sqrt {15}}{2}\right )}{5}-\cos \left (\frac {t \sqrt {15}}{2}\right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {t}{2}} \left (-\frac {4 \sqrt {15}\, \sin \left (\frac {t \sqrt {15}}{2}\right )}{5}-4 \cos \left (\frac {t \sqrt {15}}{2}\right )\right )}{4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 92
DSolve[{x'[t]==2*y[t],y'[t]==-2*x[t]-1*y[t]},{x[0]==-1,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{5} e^{-t/2} \left (\sqrt {15} \sin \left (\frac {\sqrt {15} t}{2}\right )-5 \cos \left (\frac {\sqrt {15} t}{2}\right )\right ) \\ y(t)\to \frac {1}{5} e^{-t/2} \left (\sqrt {15} \sin \left (\frac {\sqrt {15} t}{2}\right )+5 \cos \left (\frac {\sqrt {15} t}{2}\right )\right ) \\ \end{align*}