y" + 25y = cos(5t), y(0) = 3, y'(0) = 9. a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). b. Solve your equation for Y(s). Y(s) = L {y(t)} = help (formulas) c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). y(t) =

Problem #3: Solve the following initial value problem. x"+9xy' + 20y = 0, Problem #3: y(1) = 7, y'(1) = 8 Enter your answer as a symbolic function of x, as in these examples Do not include 'y = 'in your answer.

Justify and analyze the model proposed in equation (2), covering in particular the following issues: an analysis of the validity of equation (2) as a model for the fish population in a fish farm; an analysis of whether, based on model equation (2), we may expect a stable fish population from which harvesting could take place; and, if so, an analysis of how large an initial population of fish will be required to obtain this stable population and the length of time required for the stable population to be established (numerical solver recommended).