Math 119 A, Midterm


ODEs and Dynamical Systems

Solve the 2x2 ODE systems $\dot x = Ax$ by finding the eigenvalues and eigenvectors and exponentiating the matrix P^-1AP,

1.

$$A = {\left(\begin{array}{cc} -4 &5\\ -1 &2 \end{array}\right)}.$$

2.

$$A = {\left(\begin{array}{cc} 1 &-4\\ 1 &1 \end{array}\right)}.$$

3.

$$A = {\left(\begin{array}{cc} -1 &1\\ 0 &-1 \end{array}\right)}.$$

Draw the phase portraits for the three ODEs above and classify the flow as sinks, sources, centers, etc.

4.

$$A = {\left(\begin{array}{cc} -4 &5\\ -1 &2 \end{array}\right)}.$$

5.

$$A = {\left(\begin{array}{cc} 1 &-4\\ 1 &1 \end{array}\right)}.$$

6.

$$A = {\left(\begin{array}{cc} -1 &1\\ 0 &-1 \end{array}\right)}.$$

7.

Solve the IVP,

$$\dot x = Ax, \: \: x(0) = x_o.$$

with

$$A = {\left(\begin{array}{cccc} -1 &-2 &1 &0 \\ 2 &-1 &0 &1 \\ 0 &0 &-1 &-2\\ 0 &0 &2 &-1 \end{array}\right)}.$$

What happens to the solutions x(t) as $t \to \infty$?