(Solved) : Part Ii Let S Consider Second Example Dx Dt Xy Dy Dt Y X 2 4 10 Points Show Equilibrium Or Q36921630 . . . .

Part II. Now let’s consider our second example:

Dx/dt = xy

Dy/dt = −y + x^2

(4)

a)(10 points) Show that the only equilibrium is the origin withone eigenvalue 0 and the other −1 for the Jacobian. Determine theequations of the lines determined by the eigenvectors correspondingto these two eigenvalues. Denote them by Ec and Es. Also find aninvariant line and sketch the corresponding phase line.

In order to understand the behaviour of the solutions near theorigin, we will try to determine a curve y = h(x) (in fact anapproximation of this curve), curve passing through the origin andtangent at the origin to Ec such