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First Order Differential Equations
In this chapter we will look at solving first order
differential equations. The most general first order differential equation can
be written as,
As we will see in this chapter there is no general formula
for the solution to (1).
What we will do instead is look at several special cases and see how to solve
those. We will also look at some of the theory behind first order differential
equations as well as some applications of first order differential equations.
Below is a list of the topics discussed in this chapter.
Linear
Equations 
Identifying and solving linear first order
differential equations.
Separable
Equations Identifying and solving separable first order
differential equations. We’ll also start
looking at finding the interval of validity from the solution to a differential
equation.
Exact
Equations Identifying and solving exact differential
equations. We’ll do a few more interval
of validity problems here as well.
Bernoulli
Differential Equations In this section we’ll see how to solve the
Bernoulli Differential Equation. This
section will also introduce the idea of using a substitution to help us solve
differential equations.
Substitutions
We’ll pick up where the last section left off
and take a look at a couple of other substitutions that can be used to solve
some differential equations that we couldn’t otherwise solve.
Intervals of
Validity Here we will give an in-depth look at
intervals of validity as well as an answer to the existence and uniqueness
question for first order differential equations.
Modeling
with First Order Differential Equations Using first order differential equations to
model physical situations. The section
will show some very real applications of first order differential equations.
Equilibrium
Solutions We will look at the behavior of equilibrium
solutions and autonomous differential equations.
Euler’s
Method In this section we’ll take a brief look at a
method for approximating solutions to differential equations.