Paul ~ Do you still need help with this, or did you get it???

 

Well, we turned that part in today, so it won't help my grade any.. but I'd like to know how to do it if you know.

 

I took diff eq in college, and it was by far the hardest class I ever took. It didn't help that the professor was a FOB chinese with a passing familiarity with the English language. I swear I heard him sneak in a few Mandarin words in every lecture.

edit: good luck!

 

I'll see if I can do this without writing a danged book,,,

To keep things straight in my head, I have to solve the equation and then find the series, if this is at all possible!! As you know, many Diff, EQ's are impossible to solve. So, we take a 'guess' and keep going!!

y' + 2xy =0
y' = -2xy
dy/y = -2x dx
ln y = -x^(2)
y = Ce^(-x^2)

We know the series for e^(x) is ~

1 + x + x^(2)/2! + x^(3)/3! +x^(4)/4!...

So, the series for e^(-x^2) is (replace every x with -x^2) ~

1 + (-x^2) + (-x^2)^2/2! + (-x^2)^3/3!....

= 1 -x^2 + x^4/2! - x^6/3! + x^8/4!...

You could keep going, but you can start to see a bit of a pattern, that these are going to be all even terms.

So, now I go back to the original equation and take my best 'guess' as to what the solution may look like, because you know that a good guess is all you're gonna get anyways!!

y' + 2xy = 0

y = a + bx + cx^2 + dx^3
y' = b + 2cx = 3dx^2

b + 2cx + 2xa +2bx^2 + 2cx^3 + dx^3 = 0

To make this true, these must also be true,,,

b = 0
2c + 2a = 0

Therefore,

y = [sum of CnX^n from n=0 to infinity]
y' = [sum of nCnX^(n-1) from n=0 to infinity]

Now combine them, you got here,,,

[sum of nCnX^(n-1) from n=0 to infinity] + 2x [sum of CnX^n from n=0 to infinity] = 0

[sum of nCnX^(n-1) from n=0 to infinity] + [sum of 2CnX^(n+1) from n=0 to infinity] = 0


Now, I start plugging in values for n and look for a pattern,,,

n = 0 2C(sub)0(X)
n = 1 C(sub)1 + 2C(sub)1x^2
n = 2 2C(sub)2X + 2C(sub)2X^3
n = 3 3C(sub)3X^2 + 2C(sub)3X^4
n = 4 4C(sub)4X^3 + 2C(sub)4X^5

Now, I figure out which of these values plugged back into the above equation will make this mess equal to zero,,,

C(sub)1 = 0
2C(sub)1 + 2C(sub)2 = 0
2C(sub)1 + 3C(sub)3 = 0
2C(sub)2 + 4C(sub)4 = 0

Can you see the pattern yet?? The odd values of n make the equation equal zero. And I'm not surprised because if you look at the original series we came up with, they were all even terms. What all this junk really tells us is that you have a homogeneous function, which basically states that ~

f(y',y) = 0

f(x) = f(0) + f'(0)x/1 + f"(0)x^2/2!...

Which also says,,,

f(x) = a + bx + cx^2

There, clear as mud!!!! Ha! Ha! Ha!! All kidding aside, I hope this helps somewhat,,, It may have been easier to fax over my notes so you had something better to look at. Andy is right, this is a very difficult class, indeed!! I need a beer!!

 

Now I know why bunny is working on those spinny things and I'm sitting here with a stupid ? over my head.

No wonder she never replied when I mentioned I was lousy at math. LOL!!

You mean to tell me that all that bs actually MEANS something.... worth remembering?

FL55 is not a space age personality. If that stuff is common... I must be Cro-Magnon.

Holy Cow. You math people!!!!

 

Yes, it means something, but who the heck remembered?!?!? I've been pouring over old notes for three days!! Ha! Ha! Ha!!

 

Well, lady.... ya impressed the heck outta me. All that stuff looked like a dusty chicken coop & a lotta hen scratchin'!!

I failed simple Algebra, with flying colors. Wierd..but math is my only area, in testing, that I score low. I've designed and built things that I've needed... using supposed algebraic formulas... not knowing that's what they were. LOL.

 

With the exception of helping out Paul, if I can, I don't see that I will ever need to use Diff. EQ's ever again. I'm not shooting to be an engineer,,, I just like playing around on those spinny things!! Ha! Ha! Ha!!

 

Wow, thanks for the help. I haven't had time to look it over completely with this paper I've been writing, but thats impressive work! Part of my problem with the class is the teacher never tells us what we're doing, he just starts spouting off names and formulas with no purpose for them, so its hard to remember.

thanks again, i'll probably check it out again later.

ps: I have some dynamics hw that needs to be done too, i'll give you a dollar..

 

I'll take it!! Ha! Ha!


Just try to remember, they give you the answer, you are trying to find the equation that goes with it,,, I'm not real good at going backwards, so it takes me forever!!