Technique #2 : The "Annihilator" Method:

An alternative to the first technique would be theAnnihilatormethod. As it's name foretells, we annihilate the non-homogeneous term and make the equation homogeneous.

To use the Annihilator technique, you must rewrite the DE usingDifferential Operator Notation:

Note that we "factor" out a y (I use this termvery loosely; you really can't factor out the y, but as you will see, it will work out to our advantage )

You can apply the annihilator to any of the following families of functions that can be:

1.

2.

3.

4.

:

The Annihilators

will annihilate .

I will leave it for you to prove the other two:

will annihilate .

will annihilate and

.

Example 17:

Solve.

Solving the homogeneous equation , we see that . Thus the complimentary solution is .

Now, to apply the annihilator to the DE, we need to rewrite it in differential form:

noting that , we have the DE:

Let us determine the proper annihilator:

. This may pose a problem: we have two different annihilators! so which one do we apply to the DE? the answer isboth. There is a theorem that states something like the following:

If there are two functions and their annihilators are respectively,

then the product of the two annihilators will annihilate .

Thus, will annihilate .

Applying the newly found annihilator to both sides we get:

Now rewrite the equation so we get the characteristic equation:

Solving for r, we get with multiplicity 2 and with multiplicity 2.

Note that 2 of the r values were values used to determine the complimentary solution! Thus, the general solution will be:

Now find the coefficients (which I will denote by A and B, respectively).

Substituting these values into the DE, we get:

Comparing the coefficients, we get:

This gives us and

Therefore the general solution is:

Example 18:

Solve(WARNING!! THIS IS A VERY TEDIOUS PROBLEM TO SOLVE!!! )

Here's the easy part (solve the homogeneous equation):

Thus the complimentary solution will be

The next part isn't that bad (solving the non-homogeneous equation):

Now we can find the annihilator:

will annihilate .

Applying the annihilator to both sides, we get:

Converting it to the characteristic equation, we have:

with multiplicity three (note that one of these roots form the complementary solution)

Now here comes the nasty part: find

I'll leave it for you to show that

Substituting this into the DE (), we have (...get ready...)

...well, after a decent amount of cancellations, we get:

Comparing the coefficients, we get:

Solving this system, we get

Therefore,

Therefore, the general solution is:

I will discuss Variation of Parameters tomorrow (hopefully)