Internal problem ID [10520]
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 }\left (t \right )&=-x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 45
dsolve([diff(x(t),t)=-x(t)-2*y(t),diff(y(t),t)=2*x(t)-y(t)],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-t} \left (\cos \left (2 t \right ) c_{1} -\sin \left (2 t \right ) c_{2} \right ) \] \[ y \left (t \right ) = {\mathrm e}^{-t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (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*}