9) Consider the initial value problem: Y” + 4y = tu(t-1) y(0) = 1, y’(0) = 0 (9a) Take the Laplace Transform of both sides of the equation and solve for Y(s)

9) Consider the initial value problem:

Y” + 4y = tu(t-1)

y(0) = 1, y’(0) = 0

(9a) Take the Laplace Transform of both sides of the equation and solve for Y(s).

(9b) Take the inverse Laplace transform of Y(s) to find y(t).

11) The system of ODE’s given by:

Has eigenvalues l = 4, 4, 4.

(11a) Find the algebraic multiplicity k, geometric multiplicity p, and the defect d of the eigenvalue l = 4.

(11b) Find the general solution to this system of ODEs.

12) Consider the nonlinear system of equations:

X’ = 1 – x + y

Y’ = y + 2x²

There are these parts to this questions: (a), (b) and (c). Be sure to answer each part.

(12a) find the critical point(s) of system.

(12b) Classify the type and stability of the critical points.

(12c) Sketch a plot of the phase portrait near each critical point using the information you determined in parts (a) and (b). This can be a rough sketch and doesn’t need to be exact. Be sure label your axes and identify any interesting points.