MATH 3328 Differential Equations

Problem Set 1

Due February 4, 2014

  1. Suppose a species of fish in a particular lake has a population that is modeled by the logistic model with growth rate and carrying capacity. Let

a)Write the equation that models this fish population.

b)Adjust the model to account for 50 fish harvested per year. What does this model predict for the long-term behavior of the fish population? Back up your answer with a qualitative analysis.

c)Adjust the model to account one-third of the fish population being harvested each year. What does this model predict for the long term behavior of the fish population? Once again, back up your answer with a qualitative analysis.

  1. Solve the initial value problem:
  2. Let

a)Make a rough sketch of the slope field without using any technology.

b)Using your drawing, sketch the graphs of the solutions with the initial conditions, , , , and .

c)Confirm your answer using HPGSolver (Attach print out of graph).

  1. Let

a)Use Euler’s method with the step size to approximate the solution to the given initial value problem over the time interval specified. Include a table of the approximate values and a sketch of the graph of approximate solutions.

b)Do the qualitative analysis of the solution of the initial value problem above and compare it to your results from Euler’s method.

c)Explain any differences. If there are differences, what went wrong?

  1. Joey decides to take all the money in his piggy back and open an ING-direct savings account. His initial deposit is $550 and the interest rate on the account is 3% per year. Joey’s Uncle Steven (referred to as Uncle Tee Tee) gives Joey $20 per week which he deposits into his savings account.

a)Assume the weekly deposits are close to continuous deposits so we can reasonably approximate Joey’s balance using a differential equation. Write an initial-value problem for his balance over time.

b)Joey started saving when he was 1year old (he is a planner). How much money will he have in his account for a car when he turns seventeen?

  1. Section 1.2: 42
  2. Section 1.3: 14