Solve the following initial value problem:

y'' - 4y' + 13y = exp(-2t)sin(t), y(0) = 1, y'(0) = 0.

a) Write down the expression for the Laplace Transform Y(s) of the solution. Output using a comment in your script.

b) Obtain the solution y(t) using inverse Laplace transform in Symbolic Toolbox. Output using a comment in your script.

c) Numerically solve the IVP using ode45. Plot y(t) and y'(t) in the time interval [0, 4] in the left two panes and the phase plot y'(t) vs. y(t) in the right 2 panes of a single plot figure. Annotate all plots. Call these solutions the ODE-y(t) and ODE-y'(t).

d) Use the solution obtained in part (a) and matlabFunction() to create a function for y and y'. Output using a comment in your script.

e) Overlay the component plots y(t) and y'(t) for analytic solutions from part (d) on the same plots as the numeric solutions of part (c). Overlay the phase plots y'(t) vs. y(t) for analytic solutions from part (d) on the same plots as the numeric solutions of part (c). Use legend to differentiate plots. Annotate all parts.