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

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:

e^2x can be annihilated to D - 2.

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

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 e^{2x + 1}?

Any help is appreciated. ;)

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

1) Differentiate Original Function
eg:
F=e(2x+1)
DF=2e(2x+1)

2) Relate Differential to Original Function
eg:
DF=2e(2x+1)
DF=2F

3) Solve
eg:
DF=2F
DF-2F=0
F(D-2)=0

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.

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!

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.

Yeah, I was looking at some of the examples later in my homework assignment that deal. For example: 6sin(x), x²e^x + 5, and e^xsin(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

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

(Fx) + D^2(Fx)=0
(Fx)(1+D^2)=0

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

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. ;)

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

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

where:

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...