- #1
stunner5000pt
- 1,461
- 2
For this linear system with complex eigenvalues
a) find the eigenvalues
b) determine whether the origin is a spiral source, sink or center
c) Determine the direction of oscillations, clockwise or anticlockwise
[tex] \frac{dY}{dt} = \left(\begin{array}{cc}0&2\\-2&0\end{array}\right) Y [/tex] with initial conditions [tex] Y_{0} = (1,0) [/tex]
i foudn the eigenvalues to be
[tex] \lambda = \pm i \sqrt{2} [/tex] which would make it a center
also the eigenvectors
[tex] \left(\begin{array}{cc}0&2\\-2&0\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right) = \pm i \sqrt{2} \left(\begin{array}{cc}x\\y\end{array}\right) [/tex] i computed to be
[tex] V_{1} = \left(\begin{array}{cc}i\sqrt{2}\\1\end{array}\right) [/tex]
and [tex] V_{1} = -V_{2} [/tex]
i feel i made a mistake in finding the eigenvectors
also what would be the direction of the oscillations then?? Do i solve the Initial value problem to get hte direction of the oscillations??
a) find the eigenvalues
b) determine whether the origin is a spiral source, sink or center
c) Determine the direction of oscillations, clockwise or anticlockwise
[tex] \frac{dY}{dt} = \left(\begin{array}{cc}0&2\\-2&0\end{array}\right) Y [/tex] with initial conditions [tex] Y_{0} = (1,0) [/tex]
i foudn the eigenvalues to be
[tex] \lambda = \pm i \sqrt{2} [/tex] which would make it a center
also the eigenvectors
[tex] \left(\begin{array}{cc}0&2\\-2&0\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right) = \pm i \sqrt{2} \left(\begin{array}{cc}x\\y\end{array}\right) [/tex] i computed to be
[tex] V_{1} = \left(\begin{array}{cc}i\sqrt{2}\\1\end{array}\right) [/tex]
and [tex] V_{1} = -V_{2} [/tex]
i feel i made a mistake in finding the eigenvectors
also what would be the direction of the oscillations then?? Do i solve the Initial value problem to get hte direction of the oscillations??