Calc 2 Lecture Notes Section 7.1 Page 2 of 5

Chapter 7: First-Order Differential Equations

In this chapter, we will look at a new way of defining a function using a new type of calculus equation. This is going to be analogous to what we do in algebra when we write and solve an equation: we know what kind of calculation to make with a number, and we know what we want the answer of that calculation to be, it’s just that we don’t know what number to use in the calculation, so we write an equation to figure it out.

Example: I am planning a family getaway to Great Wolf Lodge in the Dells for spring break. It will cost $250 for waterpark passes, and $129 a night for the room. If the room tax is 12%, and the passes are taxed at the usual sales tax rate of 5.5%, how many nights can we stay if we want to spend $700 or less on the hotel and waterpark?

Let n be the number of nights we stay…

($250)(1.055) + (n×$129)(1.12) £ $700

è n £ 3.019

è 3 nights is the most we could stay

The analogy to be made is that in a differential equation, we know what kind of equation the derivative of a function should obey, so we solve that equation to find the function.

Example: The population of a bacterial colony increases at a rate that is proportional to how many bacteria are present. If the colony grows at a rate of 6% per hour and has 100 bacteria after 2 hours, what function models the population as a function of time?

Let t represent time (measured in hours), and p(t) be the function for how many bacteria are present at time t.

Rate of increase: … notice: the units of this expression are number of bacteria per hour

Increase is proportional to how many bacteria are present:

è

Notice: this is the correct solution because , which is how many bacteria the colony had at 2 hours, and this function also obeys the defining differential equation:

Section 7.1: Modeling with Differential Equations

Big idea: Many real-world situations can be modeled with functions that are defined in terms of a differential equation. We will learn how to solve the simplest of these equations by re-arranging them and integrating both sides, and how to apply known solutions.

Big skill:. You should be able to apply the solutions of the differential equations presented in this section to concrete real-world examples.

A differential equation involves derivatives of a function in combination with the function and the independent variable of the function.

A first-order differential equation involves only the first derivative of the function.

Examples of first-order differential equations include , , and .

To solve first-order differential equations, isolate all terms with the function on one side and all terms with constants and the independent variable on the other side, then integrate both sides of the equation. The constant of integration that arises will be determined by other information given about the system. This information is called an initial condition. Here is how this all works for the problem on the previous page:

In this section, we examine only two differential equations.
One of them is the first-order differential equation obeyed by the function, which represents situations with exponential growth or decay (this applies to the example we just did for population growth, and to radioactive decay and to continuously compounded interest)
The other is the first-order differential equation obeyed by , which represents situations with exponential growth or decay to or from a non-zero asymptote (this applies to Newton’s Law of Cooling).

Exponential growth/decay:
/ Exponential growth/decay with a non-zero asymptote:

First Order Differential Equation:
/ First Order Differential Equation:

Meaning of the constants:
A: initial value of the function at t0.
t0: initial time at which A is specified
k: proportional rate of growth or decay / Meaning of the constants:
A: initial difference between the value of the function and R at t0.
t0: initial time at which A is specified
k: proportional rate decay (or growth)
R: asymptotic value of the function as the decay progresses.
Picture for k > 0:

Picture for k < 0:
/ Picture for k < 0:

Half-life constant for exponential decay (recall that k < 0): / Doubling time for exponential growth:

Practice:

  1. The population of the earth was 3.91 billion people in January of 1970, and 6.03 billion people in January of 2000. What will the population be in 2026, when you instructor retires (if he is not dead by then)?
  1. The half-life of morphine in a person’s body is 3 hours. If there is initially 0.4 mg of morphine in the bloodstream, when does the amount drop below 0.01 mg? By what percentage has the has the amount of morphine decreased after one day?
  1. A cold drink is poured from a can at 50° F. After 2 minutes of sitting in a 70° F room, its temperature has risen to 56° F. Find the drink’s temperature at any time t.
  1. If you invest $100 a month this year in an IRA that averages an interest rate of 10% a year, how much will your investment be worth in 2046, when you retire?