Diff equations

Name:

• I will be checking for organization, conceptual understanding, and proper mathematical communication, as well as completion of the problems.

• Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are doing. • Use correct mathematical notation. • You may work with your classmates. However, please submit your own work! • Please use this sheet ONLY as a cover page. Work on a separate sheet of paper.

1. Bernoulli Equations Differential equations of the form

dy

dt + g(t)y = b(t)yn (1)

where n = 0, 1, 2, . . . and both g(t) and b(t) are continuous functions (of t) are called Bernoulli Equations. (Notice that if either n = 0 or n = 1 then the equation should already be familiar to you and you know exactly how to solve in these cases. Therefore, we are going to be looking at solutions for values of n other than these two for the rest of the problem.)

(a) (5 points) Show that (1) can be converted into

dv

dt + (1 − n)g(t)v = (1 − n)b(t) (2)

by using the substitution v = y1−n . [Hint: Eliminate both y and its derivative (i.e., dy/dt) by using the

suggested substitution. Also, you are going to have to time-differentiate both sides of the equation v = y1−n

implicitly since both v and y are functions of t.] (Note: Equation (2) is a linear differential equation that you can solve for v by the Integrating Factors Method and once you have that in hand you can also get the solution to the original differential equation by plugging v back into v = y1−n equation and solving for y.)

(b) (5 points) Solve the following IVP. [Hint: use part a!]

dy

dt +

1

t y = ty2, y(1) = 2.

2. Chemicals in a Pond Consider a pond that initially contains 10,000 gallons of fresh water. Stream water containing an undesirable chemical flows into the pond at the rate of 5,000 gallons/year, and the mixture in the pond flows out through an overflow culvert at the same rate. The concentration of chemical in the incoming water varies periodically with time t (measured in years) according to the expression ‘sin(2t) + 1’ (in grams/gallon). Assume that the chemical is uniformly distributed throughout the pond.

(a) (4 points) Construct a mathematical model of this flow process. That is, state the corresponding IVP for the amount of chemicals in the pond. [Hint: you should end up with a constant coefficient linear differential equation.]

(c) (1 point) Discuss the adequacy of the model itself for this problem. That is, discuss some of the simplifying “hidden” assumptions made by the problem.

1

(b) (5 points) Solve the IVP in part (a) and plot the solution (you may use a graphing software for this). Also, use the solution obtained to predict the amount of chemicals in the pond when t = 6 years. Round your answer reasonably. [Hint: See HW4 Section 1.8 Problems 10, 20 to help inspire a particular solution guess - you may have three undetermined coefficients!]