|Jul11-12, 10:40 AM||#18|
Laplace Transform of...
If the Laplace transform of u(x) is U(s), what is the Laplace transform of u''(x) ?
|Jul11-12, 12:41 PM||#19|
Oh, the question wasn't clear. My bad.
The problem is what constants should I use. Say for example, y(0) = 1.
the equation is in terms of u, u" - 0.625u = 0
The laplace transform is s2U - su(0) - u'(0) - 0.625U = 0.
What would be my u(0) and u'(0)? Thanks. :D
|Jul11-12, 03:50 PM||#20|
If u(0) and u'(0) are not specified in the wording of the problem, then consider them as paramaters. For example, let u(0)=C1 and u'(0)=C2. They will appear into the general solution of the ODE. Since it is a second order ODE, the general solution must include two arbitrary constants anyway.
|Jul12-12, 02:32 PM||#21|
I have one last question about Laplace.
There's this convolution integral that I read, so it's integral from 0 to t of f(t-v)g(v)dv.
Solving this with some elementary algebraic maneuvers:
(2s^2 - 16) / (s^3 - 16s) = s^2 - 16 / s(s^2 - 16) + s^2 / s(s^2 - 16).
= 1 / s + s / (s^2 - 16) = 1 + cosh4t.
Using that integral; transforming the equation into:
= 2s^2 / s(s^2 - 16) - 16/s(s^2 - 16)
= 2s / (s^2 - 16) - 16/s * 1/(s^2 - 16)
= 2cosh4t - 16/4s * 4/(s^2 - 16)
= 2cosh4t - 4/s * 4/(s^2 - 16)
Using the second term;
F(s) = 4/s
G(s) = sinh4t
f(t-v) = 4
g(v) = 1/4 cosh4t
-int(0,t,4*cosh4t dt) = 4/4 cosh 4t = -cosh4t
= 2cosh4t - cosh4t = cosh4t.
The algebraic maneuver had 1 + cosh4t while this integral gave me just the cosh4t.
Where did the 1 go? Is it a constant generated from integrating? Isn't it supposed to be accounted already when the DE was transformed to the S domain (the y(0) and y'(0))?
|Jul14-12, 05:38 AM||#22|
Bump for my last question :(
|Jul14-12, 05:59 AM||#23|
For (2s^2 - 16) / (s^3 - 16s) = s^2 - 16 / s(s^2 - 16) + s^2 / s(s^2 - 16).
= 1 / s + s / (s^2 - 16) = 1 + cosh4t, I think you have made a mistake.
I get (2s^2 - 16) / s(s^2 - 16)
= (s^2 + s^2 - 16) / s(s^2 - 16)
= s^2 / s(s^2-16) + 1/s
The above will give you a different answer than your expansion.
|Jul18-12, 06:39 AM||#24|
But i think we got the same :\
I mean, taking yours:
= s^2 / s(s^2 - 16) + 1/s ;cancelling extra s
= s / (s^ - 16) + 1 / s = cosh4t + 1
|Jul18-12, 06:59 AM||#25|
Haha yeah you're spot on! I guess if thats right the next thing to look at is the convolution.
So in the convolution you wrote:
4*1/4(cosh4t - 1] = cosh4t - 1
Now 2cosh4t - (cosh4t - 1) = cosh4t + 1.
|Jul20-12, 02:48 AM||#26|
Oh, that was a typo. I was sleepy when typing that lol, sorry.
That's where i'm having issues. Where did the -1 come from? :(
i mean, f(t-v) = 4, g(v) = sinh4v
-int(0,t, 4 sinh4v dV) = -4/4 sinh 4t, right? where did the -1 come from? :S
|Jul20-12, 03:04 AM||#27|
argh, I'm thinking too much. I see now where it came from. Thanks for answering.
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