Query 31

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course MTH 279

4/26 1pm

Query 31 Differential Equations*********************************************

Question: Using, if necessary, the table in your text, find the Laplace transform of e^(3 t - 3) * h(t - 1), where h(t) is the Heaviside function.

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Your solution:

e^(-3-(s-3))/(s-3) , s > 3

The first thing I did was pull the constant e^-3 out of the equation. I then looked at what I had left and decided that my e^3t part would shift the arguement of my transform I then found the general trasform of of h(t-1) to be e^-s/s, which I applied the shift of (s-3) to.

@&

e^(-3-(s-3))/(s-3) = e^-s / (s - 3),

since -3 - (s - 3) = -s

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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: OK

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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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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: Using, if necessary, the table in your text, find the inverse Laplace transform of 10 / (s^2 + 25) + 4 / (s - 3).

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Your solution:

f(t) = 2sin5t + 4e^3t * h(t)

I used the fact that these were linear combos to find the inverse of each piece individually.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

THe only thing I am unsure about is how to handle the constants in this situation (like the 2 and 4 I came up with) I assumed I could just pull them out front and work with the rest of the equation, since this is what I normally do when taking derivatives/integrals, and the Laplace transform just seems to be an extension of that.

@&

You do have a constant-multiple rule throughout, whether transforming a function or finding an inverse transform.

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@&

This is a direct result of the constant-multiple rule for integration and differentiation.

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Self-critique rating: 3

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Question: Using, if necessary, the table in your text, find the inverse Laplace transform of e^(-2 s) / (s - 9).

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Your solution:

f(t) =e^9t * h(t-2)

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: Using, if necessary, the table in your text, find the inverse Laplace transform of 1 / (s + 1)^3

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Your solution:

(1/2)e^-t * t^2

if t=2, then I needed to turn 2! into 1, hence the extra 1/2 on the outside. Besides that, I knew my shift came from my (s + 1), which corresponds to my e^-t term.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: Using, if necessary, the table in your text, find the inverse Laplace transform of (2 s - 3) / (s^2 - 3 s + 2).

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Your solution:

f(t) = e^2t * h(t) + e^t * h(t)

I used partial fractions to break this one up into 1/(s-2) and 1/(s-t) and found the inverse transforms of these pieces individually since they were linear combos.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

I'm rather certain I did my partial fractions right, but I didn't know if partial fractions was the right way to do it.

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Partial fractions is about your only option here.

Good work.

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Self-critique rating: 3

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