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Initially a 200 gallon tank is filled with pure water. At time [pic] a salt concentration with 3 pounds of salt per gallon is added to the container at the rate of 4 gallons per minute, and the well-stirred mixture is drained from the container at the same rate.

a. Find the number of pounds of salt in the container as a function of time.

[pic][pic][pic][pic]

This is separable: [pic]

Initially, [pic] So, [pic]

b. How many minutes does it take for the concentration to reach 2 pounds per gallon?

Note that 2 lb/gal means that [pic]lb. So, [pic]min.

c. What does the concentration in the container approach for large values of time? Does this agree with your intuition?

Let [pic] Then [pic]lb and the concentration [pic]

This is expected, as this is the concentration of the incoming mixture.

d. Assuming that the tank holds much more than 200 gallons, and everything is the same except that the mixture is drained at 3 gallons per minute, what would the answers to parts a and b become?

This changes the problem because the volume increase at (4-3) gal/min. So, we replace the volume 200 gal by [pic] gal ant its rate is 3 gal/min instead of 4 gal/min. Thus, [pic]. This is a first order equation.

The integrating factor is [pic]

The new equation becomes: [pic]

Integrating, we obtain [pic]

For [pic] Plotting the solution, one finds that the level reaches [pic] around [pic] min.

1. The differential equation governing the velocity of an object of mass m subject to air resistance that is proportional to the square of the velocity is given by [pic] Solve the equation and determine the limiting velocity as [pic] Rewrite this separable equation: [pic]

Integrating, we have [pic] for [pic]

Thus, [pic] where [pic]

You can solve for v: [pic] and then clean up the constants.

As [pic]

Note: There is a much cleaner answer obtainable with a hyperbolic function substitution at several places in the derivation. See the next homework.

2. Consider the following equations. Determine the equilibrium solutions and sketch the behavior of the solutions for all initial conditions. Stable equilibrium solutions are those in which nearby solutions approach equilibrium. Unstable equilibrium solutions are those in which nearby solutions go away from equilibrium. Classify your solutions in each case.

a. [pic] Equilibrium Solutions: [pic]

[pic]

b. [pic] Equilibrium Solutions: [pic]

[pic]

c. [pic] Equilibrium Solutions: [pic]

[pic]

3. Consider a logistically changing fish population with constant harvesting/ restocking, given by the initial value problem [pic] For various values of H, the general population solutions may behave differently. Examine the equilibrium solutions for the different values and describe what you find.

Equilibrium solutions: [pic] The number of equilibrium solutions depends upon the nature of these solutions.

1. [pic] Two equilibrium solutions

2. [pic] One equilibrium solution

3. [pic] No equilibrium solutions.

Pick sample H values for each case to see how other solutions behave like in Prob 3.

|[pic] |[pic] |[pic] |

4. You make two gallons of chili for a party. The recipe calls for two teaspoons of hot sauce per gallon, but you had accidentally put in two tablespoons per gallon. You decide to feed your guests the chili anyway. Assume that the guests take 1 cup/min of chili and you replace what was taken with beans and tomatoes without any hot sauce. [1 gal = 16 cups and 1 Tb = 3 tsp.]

a. Write down the differential equation and initial condition for the amount of hot sauce as a function of time in this mixture-type problem.

[pic]

IC. Two Tb per gallon for two gallons implies that [pic].

b. Solve this initial value problem.

This is a simple exponential model: [pic]

c. How long will it take to get the chili back to the recipe’s suggested concentration?

This is when the amount is 4 tsp, since there are two gallons. So, [pic] min.

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