Differential Equations: Undetermined Coefficients (Annihilator Method) (in Off-topic)
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. ;)
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
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.
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!
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.
ok i see you had the F(x) out front muliplying through the D-2....didnt see that far down!
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.
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. :)
for the 6sin(x) wouldnt it be something like F(x) + D^2 (F(x)) = 0 ?
6sin(x) - 6 sin(x) = 0
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 :)
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. ;)
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
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. ;)
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
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.
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>