Two tanks, A and B, contain 500 liters each of saline solution with a concentra- tion of 10 grams/liter. The tanks are connected by a pipe, and there is a pipe leading into tank A, and a pipe leading out of tank B, as in the illustration below. A B Figure 1: Tanks for problem 1 At time t = 0, pure water begins to flow into tank A at a rate of 20 liters/min, the well-mixed solution flows from tank A to tank B at the same rate, and the well-mixed solution flows out of tank B at the same rate. (a) (b) Find initial value problems that model the quantities QA (t) and QB(t) of salt, in tanks A and B respectively, t minutes after the flowing begins. Find the times tд and t when the saline solutions in tanks A and B. reach concentrations of 5 grams/liter, respectively. Comment: You can find tд by explicitly solving a simple equation. To find t, you will need to use your favorite method of finding an approximate solution. E.g., Newton's method, guessing-and-checking, etc.

The population growth of a certain species of fish in a large lake is governed by the ODE where P(t) (a) (b) (c) (d) = P' 0.25P2/3 (10 - P), = the number of fish in the lake in year t, measured in 1000s. What is the carrying capacity for this population? How do you know. Phrase your answer in terms of the equilibria of the ODE and their stability. Suppose that P(0) = 2, and the harvesting of fish begins, with H = 2 (thousands of fish) harvested every year. Use Matlab to estimate the (new) carrying capacity for this population. Explain your reasoning/methodology, and include screenshots of the corresponding Matlab commands and output. Repeat (c) with H = 3 (without the reasoning). What is the largest number of fish that can be harvested each year without killing off the population, assuming as before that P(0) = 2? Show how you reach your conclusion.

Find the general solution of the equation 4y" +8y' +5y = 0. Find the solution of the equation in (a) with y(0) = 1 and y'(0) = −1.