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APMA 0350 − HOMEWORK 2
Problem 1: (3 points, 1 point each) Do the following ODE satisfy the assumptions of the Existence-Uniqueness Theorem from lecture? Why or why not?
(a) dy dt = y t 2 + 1 y(−1) =9
(b) dy dt =y 2 (|t| + y) y(0) =2
(c) dy dt = 1 y + 1 y(1) = − 1
Problem 2: (4 points, 1 point each) Consider the equation y ′ = (y + 2)(y − 1)(y − 3)
(a) Find the equilibrium solutions
(b) Draw a bifurcation diagram (that table with signs)
(c) Draw a sketch of the equilibrium solutions and of a couple of solutions between them (that graph with the curves) (TURN PAGE)
(d) Classify the equilibrium solutions as stable/unstable/bistable.
Problem 3: (6 points, 2 points each) Solve using separation of vars
(a) (Implicit form) dy dt = tsec(y) y (e t 2 )
(b) (Remember the initial condition) dy dt = 2t + sec2 (t) 2y y(0) = − 5
(c) (Beware of hidden solutions, see last example in the notes) dy dt = −t(y − 1)2 3
Problem 4: (5 = 2 + 1 + 2 points, Application)
(a) Suppose you’re trying to model a population of bunnies. Initially there are 100 rabbits, and you notice that the growth rate is 0.08 bunnies/day. Moreover, due to limiting resources, it seems that the bunny population doesn’t exceed 1000 rabbits.
Set up a differential equations model for the number of rabbits and follow the steps from lecture to solve it. You can skip the partial fractions part if you wish.
(b) What is the limit of the solution in (a) as t → ∞ ? (TURN PAGE)
(c) For which t do we have y(t) = 900? Please find an exact formula for t, as well as an approximate value (using your calculator)
Problem 5: (2 points, Mini Theory) Show that the function y(t) whose graph is below cannot be a solution of an ODE of the form y ′ (t) = f(y(t))
Hint: Find two different values t1 and t2 on the graph with y(t1) = y(t2) but y ′ (t1) < 0 and y ′ (t2) > 0
Since the ODE is true for all t, it has to be true for t1 an t2 as well. Use that to find a contradiction
Problem 6: (4 points, 2 points each) Solve the following ODE using integrating factors (a) y ′ = 2y + t 2 e 2t (b) ty′ + 2y = sin(t) (TURN PAGE) 4 APMA 0350 − HOMEWORK 2
Problem 7: (5 = 1 + 2 + 2 points) For this problem, refer to the Chemical Tanks example from lecture Suppose a tank is filled initially (at time t = 0) with 100 gallons of fresh clean water. Water containing 10 grams/gallon of chemical pollutants enters the tank at a rate of 2 gallons/day, and the mixture in the well-stirred tank leaves the tank at the rate of 1 gallon/day.
(a) Find the amount of water W(t) as a function of time and determine the time when that amount reaches 200 gallons. In order to do this, either set up a very easy ODE for W(t) or use your intuition about the problem.
(b) Write down the differential equation that describes the total amount of pollutants P(t) in the tank and find the initial condition P(0) of this quantity at time t = 0. Again, think in terms of rate in minus rate out.
(c) Find the total amount of pollutants in the tank at the time you determined in (a) Problem 8: (7 = 1 + 1 + 2 + 2 + 1 points) For this problem, refer to the Rabbits vs Foxes example in the lecture notes, as well as the Savings example.
Our goal in this problem is to develop a model for the bunny population at Brown by taking the following factors into account:
• The initial population on Jan 1 was 50 rabbits
• On average, every rabbit has 0.1 offspring per month
• On average, 30% of the existing rabbit population dies per month
• 30 rabbits migrate into the Brown campus area per month. Using this information:
(a) Clearly identify your dependent and independent variables, including their units
(b) Derive a differential equation for the rabbit population that takes all factors listed above into account, using the trick with h small and calculating the change, similar to the Savings example in lecture.
(c) Perform a qualitative analysis: Find the equilibrium solutions, draw the bifurcation diagram, plot the equilibrium solution and at least two other solutions, and determine if the equilibrium solution is stable/unstable/bistable
(d) Solve the ODE, including the initial condition
(e) Use your solution to figure out what happens to the population of bunnies as t → ∞.
Problem 9: (4 points)
For this problem, you need to use Newton’s law of cooling which states that the rate of change of the temperature of an object is proportional to the difference between that temperature and the ambient temperature. Recall that a is proportional to b if there is k such that a = kb (TURN PAGE)
Peyam baked a Yam Pie, and he only wants to eat it once the temperature reaches 25 degrees Celsius, so it will not burn him. Initially he measured the temperature and the pie was 40 degrees Celsius. One minute later, the pie was 35 degrees. The ambient temperature of the room is 22 degrees. When should Peyam start eating the pie?
Note: It’s ok to use your calculator and approximate values here!