Hy can u plz solve these and plz do not provide me the ai generated answers provide me the solutions so that i can understand it

3. [Laplace Transforms for Solving Differential Equations] This is the heart of the post-Midterm 3 material. Here, we give you a number of examples. (Note that some of these differential equations are intentionally borrowed from the Pre-Midterm 3 Quizzes and Homework 7; feel free to check your answers using the solutions to such.) For each differential equation below: i. Find the Laplace transform Y(s) of y(t). ii. Mark the pole locations of Y(s) in the complex frequency (s) plane. iii. What does the pole structure of Y(s) tell you about the behavior of y(t)? Choose exactly one of these options. I. Grows to co without oscillation as t + co. II. Oscillates while decaying to zero as t -+ co. III. Decays to zero without oscillation as t + co. IV. Oscillates while growing without bound as t -+ co. V. Oscillates forever without decaying or growing in amplitude as t -+ co. iv. Take the inverse Laplace transform of Y(s) to obtain y(t) for t > 0. (a) y' + 5y' + 6y = 15e-8t . u(t), y(0) = 1,y'(0) =0 (b) y' + 2y = 8t - u(t) + 6e-" . u(t), y(0) =5 (c) 2y" + 50y = 20e-5t . u(t), y(0) = 1,y'(0) = 3 (d) y" + 6y' + 9y = 18t . u(t). y(0) = -2,7/(0) =7 (e) y" - 6y' + 25y = 50 - u(t), y(0) = 10,y'(0) = -20 4. For each part, write the general form of f (t) that corresponds to the poles of F(s) shown in the diagram on the next page. You will not be able to determine any leading constants, and you can use any letters you want for those. You should, however, be able to write the functional forms using known values, for example, your answers might look like: f(t) = Cie-12 + Cye-24t + Do + Dit or f(t) = Aj + Ag cos(20t) + As sin(20t)X pole @ double pole Im Im Im 6 6 A 4 4 2 2 2 Re Re Re -2 -2 -2 -4 -4 -4 -6 -6 6 4 2 0 2 4 6 -6 6 4 2 0 2 4 6 -6 4 2 0 24 6 (c) (b) (a) Im Im Im A X 2 NO 0 Re Re Re O -2 * -2 -2 4 4 -4 -6 -6 0 2 -6 -6 2 -6 4 -2 4 6 4 2 0 4 6 -6 4 2 0 2 4 6 (f) (d) (e)