Problem Set 8—Differential Equations

Purpose

This problem set reinforces your ability to solve separable and first-order linear differential equations.

Background

This problem set is based on material in sections 7.4 and 9.2 of our textbook. We discussed section 7.4 in class on March 4 and I expect to discuss section 9.2 on March 10.

Activity

Solve each of the following problems:

Problem 1

On March 4 we talked about 0^th order, 1^st order, and 2^nd order chemical reactions. For a 2^nd order reaction that consumes reactant A, we derived an equation for the concentration of A, [A], as a function of time, namely

1/[A] = kt + 1/[A]₀

where [A]₀ is the initial concentration of A, and the reaction rate is defined by

d[A]/dt = -k[A]²

0^th and 1^st order reactions are defined by the rate equations

d[A]/dt = -k
d[A]/dt = -k[A]

respectively. Solve each of these equations for the concentrations of A as functions of time.

Problem 2

Exercise 20 in section 7.4 of our text (solve dy/dx = xy + 3x - 2y - 6).

Problem 3

Exercise 24 in section 7.4 of our text (atmospheric pressure as a function of altitude).

Problem 4

Exercise 26 in section 7.4 of our text (inversion of sugar).

Problem 5

Exercise 2 in section 9.2 of our text (solve e^x dy/dx + 2e^xy = 1).

Problem 6

Exercise 26 in section 9.2 of our text (derive the equation for current in an open RL circuit).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as reading through it will help me know what to focus on in the rest of the meeting.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.

My basic expectation in grading this exercise is that your solution will show that you understand how to solve each problem, although there may be arithmetic or copying mistakes, inefficient solution methods, incorrect or irrelevant statements incidental to the solution, or similar minor mistakes. If you understand how to solve all the problems and have no minor errors, I will consider the solution to be in between “what I expect” and “surprisingly beyond expectations.” I will consider solutions to be 3/4, 1/2, 1/4, or none of what I expect according to what (rough) fraction of the problems your solution shows understanding of, although I will raise grades slightly if it is clear by the end of your grading meeting that you have come to understand things you didn’t understand when you arrived at the meeting.