The actual multiple of $\vc{u}$ and $\vc{v}$ used for the solution $\vc{y}$ depend on the initial condition $y(0)=y_0$. Given the value of the initial conditions that you calculated above, determine the solution $\vc{y}(t)$.
First, write the solution in vector form.

The first blank should be $\vc{u}$, the appropriate multiple of the eigenvector corresponding to the smaller eigenvalue, and the second blank should contain an exponential function involving the smaller eigenvalue. The third blank should be $\vc{v}$, the appropriate multiple of the eigenvector corresponding to the larger eigenvalue, and the fourth blank should contain an exponential function involving the larger eigenvalue.

Rewrite this solution in terms of the components of $\vc{y}$, which we'll call $y_1$ and $y_2$.

$y_1(t) = $

$y_2(t) = $

On the below phase plane, plot the two eigenvalues (any vectors that are multiplies of $\vc{u}$ and $\vc{v}$). To do so, drag the black points so that the green and blue vectors point in the correct direction, one for each eigenvector. (The applet automatically plots the opposite of each vector, as both directions correspond to the eigenvector.)

You can click the each vector to toggle its direction; point each vector in the direction that the solution moves along the eigenvector.

**Feedback from applet**

eigenvectors:

movement direction along eigenvector:

Lastly, solve the system for the original variables $x_1$ and $x_2$.
$x_1(t) = $

$x_2(t) = $

On the below phase plane, move the red point to the equilibrium. Then plot the two eigenvalues so that they are centered at the equilibrium. To do so, drag the black points so that the green and blue vectors point in the correct direction, one for each eigenvector. (The applet automatically plots the opposite of each vector, as both directions correspond to the eigenvector.)

You can click the each vector to toggle its direction; point each vector in the direction that the solution moves along the eigenvector.

**Feedback from applet**

eigenvectors:

equilibrium:

movement direction along eigenvector: