Differential Equations: Undetermined Coefficients (Annihilator Method) (in Off-topic)

SNK3R March 5 2006 5:18 PM EST

Just recently in my DE class, we went over this method. It's interesting, to say the least, but I have a question regarding more complex functions.

To start out, a basic function can easily be annihilated by seeing how many times it'll take the function to get the derivative equal to zero. For example:

8x3 - 5x2 + 1 can be annihilated to D4 [take the derivative four times (i.e., power + 1)].

However, when I got to exponential functions, I started not being able to "see" them quite as easily. For example:

e2x can be annihilated to D - 2.

Proven the way by my professor:
(D - 2)e2x
De2x - 2e2x = 0 [De2x is the derivative of e2x, which is 2e2x.]
If De2x = 2e2x, then we can say 2e2x - 2e2x = 0, so it's true that D - 2 is the annihilator for the function e2x.

This makes sense when it's proven, but is there a trick to see them easier than having to guess and check to see which annihilators relate to the function? For example, how would I "guess" the annihilator for the function e2x + 1?

Any help is appreciated. ;)

Sacredpeanut March 5 2006 5:35 PM EST

Not sure if this would be the "correct" method, but this is how I would do it.

1) Differentiate Original Function

2) Relate Differential to Original Function

3) Solve

In other words multiplying the original function by D-2 will annihilate it to 0. Therefore e(2x+1) can be annihilated to D-2.

SNK3R March 5 2006 5:46 PM EST

Wow, that is brilliant! That totally rocks. Thanks a bunch. Now I can actually solve for the annihilator instead of guessing!

smallpau1 - Go Blues [Lower My Fees] March 5 2006 5:48 PM EST

why is it D - 2, instead of being D - 2F ? I'm not in DE yet, im only in calc 3. But this stuff does make sense to me, what you're talking about, lol.

smallpau1 - Go Blues [Lower My Fees] March 5 2006 5:50 PM EST

ok i see you had the F(x) out front muliplying through the D-2....didnt see that far down!

Sacredpeanut March 5 2006 5:53 PM EST

No worries :)

It's easy to write an exponential function in terms of it's derivative but is alot harder with other functions so this may not be the best method for all functions.

SNK3R March 5 2006 5:56 PM EST

Yeah, I was looking at some of the examples later in my homework assignment that deal. For example: 6sin(x), x2ex + 5, and exsin(x). I guess I'll mess with them once I get to them. But this trick for exponential functions really rocks. I've already tested it on a few sample exponential functions. :)

smallpau1 - Go Blues [Lower My Fees] March 5 2006 6:00 PM EST

for the 6sin(x) wouldnt it be something like F(x) + D^2 (F(x)) = 0 ?

6sin(x) - 6 sin(x) = 0

Sacredpeanut March 5 2006 6:10 PM EST

Yes, 1+D^2 would therefore be the annihilator.

(Fx) + D^2(Fx)=0

I think I did a similar paper to you last year, so if you need any help just ask :)

SNK3R March 5 2006 6:10 PM EST

Yeah, I just did it for the problem on my homework.

D2 + 1 will annihilate 6sin(x):
6sin(x) * (D2 + 1)
D26sin(x) + 6sin(x)
-6sin(x) + 6sin(x) = 0

Good job, you'll be ready once you hit DE. ;)

Stephen March 5 2006 6:15 PM EST

Hopefully next week he'll teach you the more general variation of parameters method for those cases where an annihilator doesn't exist

SNK3R March 5 2006 6:21 PM EST

If you're referring to homogeneous linear ODE's with constant coefficients (problems that have 3 general solutions: 1 repeated solution, 2 different solutions, or 2 complex solutions?), we've already discussed that. ;) If not, I can't wait. I'm actually enjoyed Differential Equations more than calculus. ;)

smallpau1 - Go Blues [Lower My Fees] March 5 2006 6:58 PM EST

calculus sucks, period...lol.

I'm trying to do a take home test right now....

If you want, try and help me out too....lol

heres one of the questions i cant figure out:

show that dN/ds = -kT + tB

using dT/ds = kN , and dB/ds = -tN


d = derivative of
N = unit normal vector
k = kappa
T = unit tangent vector
B = Binormal vector
t = torsion

Zoglog[T] [big bucks] March 6 2006 6:27 AM EST

Differentials of parametrical trigonometrical equations are bad enough before I even reach Uni (glad to not be doing a Maths course when I get there). Implicit differentiation and rate of change isn't much better although is shortest chapter in the book before we start on vectors luckily.

AdminQBGentlemanLoser [{END}] March 6 2006 6:52 AM EST

I used to know this.


Makes me sad...
This thread is closed to new posts. However, you are welcome to reference it from a new thread; link this with the html <a href="/bboard/q-and-a-fetch-msg.tcl?msg_id=001jSS">Differential Equations: Undetermined Coefficients (Annihilator Method)</a>