Lecture 1

A survey, some examples involving money and other things, and nice pictures
called slope Fields. Reading a differential equation.
Welcome to the world of differential equations! Although you have somehow survived
life for 18 or more years without hearing much about them, differential equations happen
to describe many of the processes in the world around you. Today we are going to survey
what some of them look like.
A differential equation is an equation which contains an unknown function and one of
its unknown derivatives. Here is a differential equation.
dy
dt
= :028y
It doesn't look too exciting does it? Really it is, though. It might for example represent
your bank account, where the balance is y at a time t years after you open the account, and
the account is earning 2.8% interest. Regardless of the specific interpretation, let's see what
the equation says. Since we see the term dy=dt we can tell that y is a function of t, and that
the rate of change is a multiple, namely :028, of the value of y itself. We definitely should
always write y(t) instead of just y, and we will sometimes, but it is traditional to be sloppy.
For example, if y happens to be 2000 at a particular time t, the rate of change of y is
then :028(2000) = 56, and the units of this rate in the bank account case are dollars/year.
Thus y is increasing, whenever y is positive. What do you estimate the balance will be,
roughly, a year from now, if it is 2000 and is growing at 56 dollars/year? This is not
supposed to be a hard question. (By the way, when I ask a question, don't cheat yourself by
ignoring it. Think about it, and future things will be easier. I promise.) Later when y is,
say, 5875:33, its rate of change will be :028(5875:33) = 164:509 which is much faster. We'll
sometimes refer to y0 = :028y as the banker's equation.
Do you begin to see how you can get useful information from a differential equation
fairly easily, by just reading it carefully? One of the most important skills to learn about
differential equations is how to read them. For example in the equation
y0 = :028y