Query 30 re

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course MTH 279

I am resubmitting my queries 29 and 30 that I submitted on 4/21 because I haven't seen a reply to them on my access page. If you've been busy, I understand and it's no big deal. I just thought I'd resubmit them in case something happened and you hadn't received them. 4/26 9am

Query 30 Differential Equations

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Question: Using the definition of the Laplace transform, find the Laplace transform of f(t) = t e^(t sqrt(t)).

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Your solution:

The Laplace Transform isn't defined for this function (the integral can't be computed).

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I agree. There is no value of s for which e^(t^(3/2) - s t) does not approach infinity.

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confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating:

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Question: Using the definition of the Laplace transform, find the Laplace transform of the function f(t) defined by f(t) = 0, 0 <= 1 < 1; f(t) = t - 1, 1 <= t.

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Your solution:

L = s^-2 *e^-s

The first part of f(t) doesn't contribute to the Laplace transform. The second part, I evaluated the intergral from 1 to inf of (t-1)e^(-st) dt.

Using integration by parts, I arrived at s^-1*te^(-st) - s^-2*e^(-st) + s^-1 * e^(-st) to evaluate from 1 to inf. Doing so an simplifying led me to the above result.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

I do not know if I evaluated the limit of the expression correctly. Namely, I decided s^-1*te^(-st) approached 0 and t approached infinity, which I'm not sure is correct or not.

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Self-critique rating: 3

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Question: Using the definition of the Laplace transform, find the Laplace transform of f(t) = cos(omega t).

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Your solution:

L = (-w^2 +s)/(w^2 +s^2)

int from 0 to inf of e^(-st)cos(wt) required integration by parts 2 times.

I ended up with the expression e^(-st)[w^-1sin(wt) - sw^-2cos(wt)] all divided by (w^2 +s^2)/w^2

Evaluating the integral from 0 to inf gave me my final answer.

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Good, but

sin(omega t) = 0 when t = 0, so you don't get the -omega^2 term in the numerator.

The results is just

s / (omega^2 + s^2).

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confidence rating #$&*:

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Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: Using the definition of the Laplace transform, find the Laplace transform of f(t) = e^(3 t) sin(t).

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Your solution:

L = 1/[ 1 + (3-s)^2] when s>3

This one also required multiple integration by parts, and I came up with e^[(3-s)t] * (-cost + (3-s)sint) / (1 + (3-s)^2) to evaluate from 0 to inf.

confidence rating #$&*:

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Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): OK

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Self-critique rating: OK"

&#Good responses. See my notes and let me know if you have questions. &#

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Question: Using, if necessary, the table in your text, find the Laplace transform of e^(2 t) cos(3 t).

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Your solution:

(s-2)/[(s-2)^2 + 9] , s > 2

My e^2t part shifts the arguement of my transform form s to (s-2). The general solution to cos3t is s/(s+ 9), so applying this shift I got my final answer.

confidence rating #$&*:

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