3. The Integrating Factor

The method of the integrating factor is used when we have differential equations in the form Multiplying the equation through by theintegrating factorwe would have the equation Integrating both sides and solving for , we get:

Let us go through an easy example, and then a challenging one.

Example 5:

Solve

In order to apply the integrating factor, the coefficient ofmust be equal to 1.

Now find the integrating factor:

Multiplying through, we should get

Integrating, we find that

Imposing the initial condition we see that

Therefore, the solution to the differential equation is

Example 6:

Solve

Divide through by

Now find the integrating factor:

Apply long division to simplify the integrand:

(Verify):

Multiplying through by the integrating factor, we should get

(Verify):

Integrating both sides and then solving for , we get

Now apply the initial condition

Therefore, our particular solution will be

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4. Exact Equations

In order to use the technique to solve exact equations, the equations must be in the form:

And they must satisfy this one condition:

If this relationship is true, we'll continue on with this technique. If its not true, we will resort to 2 other possible techniques which will be discussed later.

When we go about solving this, we should make known that and that

Step one: find You can do it two ways, but I will do it this way because it's the most common way:

Step 2: Find To do this, partially differentiate with respect to

Since

Solving for we get

Integrate to find

Step 3: write solution in general form.

The general solution of an exact equation will have the form

Since the general solution will be

Example 7:

Solve

Let

and

Test for exactness:

and

They are equal, so the DE is exact.

Find

Now find

Since

Therefore,

and the general solution is

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I will be back later with more...