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) =

8. Solve the following initial value problem: (picture)

Subject:

Problem 3. A 400 gallon tank contains water into which 10 lbs of salt is dissolved. Salt water containing 3 lbs of salt per gallon is being pumped in at a rate of 4 gallons per minute, and the well mixed solution is being pumped out at the same rate. Let A(t) be the number of lbs of salt in the tank at time t in minutes. Derive the initial value problem governing A(t). Solve this IVP for A.

Problem #8: Solve the following initial value problem. y"-20y + 149y = 0, y(0) Problem #8: = 4, y'(0) = 10 Enter your answer as a symbolic function of x, as in these examples Do not include 'y=' in your answer.

(4).

A salmon population is modelled by the DTDS X+1 = 0.3 + x, • (a) ( Write down the updating function f (x) of this DTDS. (b) ( Suppose we know that X10 = 0. 5. What is the value of X11? • (c) SWrite down the equation that we need to solve in order to find the fixed points. Then solve the equation algebraically (that is, without using graphs). • (d) point. (No marks will be given if you use other methods.) Use the Stability Theorem to determine the stability of each fixed

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.

Solve the following initial value problems. У (0) %3 0, у'(0) %3D1, у" (0) %3 0 3. у" + 11у" + 39y' + 45у %3D 0 4. у" + 9у yE) = 0, y'(n) = -1 Усп) %3 0, у () - 1, у" () — о, у"(п) —1 = 0 5. у4) + 8y" +16у %3D 0 y(n) =

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).