Can someoen verify i did this differentiation right?

[mr_coffee]

Hello everyone, these problems take like 2 pages to do so i don't want to redo it all if i messed up on a derivative, can someone tell me if they get the same thing for y''?
Thanks:
http://img220.imageshack.us/img220/8417/lastscan2rf.jpg

I think I better do these problems in pencil...

 

[Fermat]

y'' is the correct derivative from y'.
But where did y' come from?
What is y?
Is y that first line on the page, with the number 2 written after it (in a circle).
If so, then your y' should have used the product rule like y'' did.

 

[mr_coffee]

Thanks for the responce Fermat, here is what y is:
http://img100.imageshack.us/img100/1671/lastscan2vi.jpg
http://img220.imageshack.us/img220/8417/lastscan2rf.jpg

So your thinking I did it wrong?

 

[Fermat]

You got the y' wrong. I'm afraid.

y = u1.f(t) + u2.g(t)

you should have, (using the product rule)

y' = u1'.f(t) + u1.f'(t)+ u2'.g(t) + u2.g'(t)

where f(t) = e^(-7t)
and g(t) = t.e^(-7t)

to get g', you have to use the product rule on g(t).

Also, I don't see hoe you got the line marked (2), from the line above it.

HOLD ON. more to come.

 

[Fermat]

I just noticed. You got a repeated root.

With repeated roots, the solution is,

y = (A + Bt)e^(rt)

where r is the repeated root.

Also, c1 and c1 are supposed to be constants of integration. Why did you replace them with functions ot time c1 = u1(t) ??

 

[mr_coffee]

Thanks for the responce fermat, I'm replacing the constants with the functions of t because I'm doing a varation of parameters technique to solve this, unless you know an easier way to solve it. Check my last thread: https://www.physicsforums.com/showthread.php?t=111140
The problem takes like 2 pages to do!

 

[Fermat]

I guess this must be just exercise in the use of variation of parameters and wronskians - yes ?
It's a lot easier to do it the usual way.

y1 = (A + Bt) e^(-7t), but you've got that haven't you.

and y2 comes out quite simply at,

y2 = 5e^t

so,

y = y1 + y2
y = (A + Bt)e^(-7t) + 5e^t
(takes about half a page)

Mind you. I've not done variation of parameters or wronskians, so I hope I'm not misleading you.

 

[mr_coffee]

Actually I'm not forced to use variation of paramters, i thought that was the only way to solve this problem because it its not homogenous!
I understand hwo You found y1, but how did u come out with y2 = 5e^t?

 

[mr_coffee]

WOw that was right and took like no work at all, if you can explain to me how you found y2 i will eat this cupcake infront of me.
http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/df/993c2e2165f062929af57210607b441.png

 

[Fermat]

finding y2 is just "standard procedure"

If the "non-homogeneous" function is say,
f(x) = a polynomial e.g f(x) = 2x² + 3x - 1, then try y2= Ax² + Bx + C
if f(x) is a trig function e.g. f(x) = sin(3x), then try y2 = Asin(3x) + Bcos(3x)
if f(x) is an exponential e.g. f(x) = 5e^(-3x), then try y2 = Ae^(-3x)

that sort of thing.

Whatever the function f(x) is, try a similar function for y2.

p.s. this standard procedue doesn't always work. Try it on y'' + 25y = 50sin(5t).

 

[mr_coffee]

hm...I c, but on the right hand size of the equation it was 320e^t
r^2+14r+49 = 320e^t,
so how did u figure 5 as being the coefficent that will work?
I would try y2 = Ae^(t) right? because its 320e^t?