In the first class, we covered what differential equations are, what makes them linear or
not linear, and why we care.
Differential equations have equal signs. Differential equations also have "fleas." (The
prime, indicating a derivative.)
The first question that one asks on seeing a differential equation is what the independent
and dependent variables are. You can tell which is the dependent variable is by looking
for the "flea" (the derivative sign).
A differential equation is an ordinary differential equation (the only kind that we will
study in Math 308) if it has only one independent variable. (There are also courses in
partial differential equations. They deal with situations where there are two independent
variables, eg., the temperature reading on a metal rod depends both on where you
measure and when you measure, ie., on two independent variables.)
The reason that one wants to know which is the dependent variable is that what is done
with the dependent variable in the equation determines if the equation is linear or
In general, if the equation is nonlinear, with very few exceptions, no one knows how to
give a formula solution. The best that one can hope for is a qualitative description of
solutions, with perhaps numeric approximations generated on a computer.
If the given differential equation is linear, there is hope that a formula solution may be
You should decide up front which kind of equation you have and what approach you are
going to take.
NSS shows in p. 5 (7) the form that a differential equation must have in order for the
equation to be linear. In words, the only things that one can do with the dependent
variable and its derivatives is to multiply them by arbitrary functions of the independent
variable and add or subtract them.
In particular, if the dependent variable is y, then the equation is nonlinear if it contains
expressions like yy', y^2, 1/y, sqrt(y''), log(y), sin(y), exp(y'), etc., which are not simply
multiplying y, y', y'', etc. by a known function of x and adding up the terms.
At the beginning of class, we practiced opening a Maple worksheet, writing mathematics,
and putting our name, UIN, and section number in a paragraph that we inserted before the
mathematics. When you work on a quiz or exam, you can always insert work in the