Internal problem ID [11538]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 4, Linear Systems. Exercises page 190
Problem number: 3(d).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }&=-x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x-y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 46
dsolve([diff(x(t),t)=-x(t)-2*y(t),diff(y(t),t)=2*x(t)-y(t)],singsol=all)
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= -{\mathrm e}^{-t} \left (c_{1} \cos \left (2 t \right )-c_{2} \sin \left (2 t \right )\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 51
DSolve[{x'[t]==-x[t]-2*y[t],y'[t]==2*x[t]-y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{-t} (c_1 \cos (2 t)-c_2 \sin (2 t)) \\ y(t)\to e^{-t} (c_2 \cos (2 t)+c_1 \sin (2 t)) \\ \end{align*}