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Math Help - Differential equations help please

  1. #1
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    Differential equations help please

    Help ! I'm working Laplace Tranforms with periodic properties & convolution and I'm not getting it. Can someone please show me how to solve these 2 problems (attached) step by step so i can understand ?


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    Quote Originally Posted by DE novice View Post
    Help ! I'm working Laplace Tranforms with periodic properties & convolution and I'm not getting it. Can someone please show me how to solve these 2 problems (attached) step by step so i can understand ?
    For the benefit of others who will look at this thread, the questions are:

    1. Solve:

    (a) y'' + 3y' + 2y = \mu_2 (t), ~ y(0) = 0, ~ y'(0) = 1.

    (b) y'' - y' + 2y = 20 \, \delta (t-3), ~ y(0) = 1, ~ y'(0) = 0.


    2. Solve (express the answer as a convolution):

    4y'' + 4y' + 17y = g(t), ~ y(0) = 0, ~ y'(0) = 1.


    As a matter of courtesy you should have done what I've done above, especially given the sort of reply you're hoping to get.

    There is a lot of routine hack work that can be done simply by applying the various formulae. So what I want to know is what have you done so far? Where exactly are you stuck?

    Please show all the work you've been able to do.

    You will also find in these forums (use the search option) several detailed solutions to similar questions.
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  3. #3
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    Is that \mu _2 in 1a) supposed to be a unit step function?

    -Dan
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    Quote Originally Posted by topsquark View Post
    Is that \mu _2 in 1a) supposed to be a unit step function?

    -Dan
    yes. From what i understand it means the function mu equals 0 when t < 2 and 1 when t >= 2
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    Here's what I have for #1a and 1b (I have it worked in the attachment as #2a & b but that's just how its numbered in my homework).....sorry its an attachment but it would be alot of stuff to key in and I'm not sure how to make it view in the message text ..still working on #2. Comments welcomed. Thanks.
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    Quote Originally Posted by DE novice View Post
    Here's what I have for #1a and 1b (I have it worked in the attachment as #2a & b but that's just how its numbered in my homework).....sorry its an attachment but it would be alot of stuff to key in and I'm not sure how to make it view in the message text ..still working on #2. Comments welcomed. Thanks.
    (2a) looks fine, but I don't see how your answer (which looks correct) follows from the third and second last lines ...... Surely you'd just apply the theorem

    LT^{-1} \left[e^{-as} F(s)\right] = f(t - a) u(t - a)

    at the fourth last line ........


    As for (2b), won't the second term of the answer contain \sinh (t - 3) rather than \cosh (t - 3) .......?


    By the way, don't bump. Replies are given when people have time.
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    Quote Originally Posted by mr fantastic View Post
    (2a) looks fine, but I don't see how your answer (which looks correct) follows from the third and second last lines ...... Surely you'd just apply the theorem

    LT^{-1} \left[e^{-as} F(s)\right] = f(t - a) u(t - a)

    at the fourth last line ........


    As for (2b), won't the second term of the answer contain \sinh (t - 3) rather than \cosh (t - 3) .......?


    By the way, don't bump. Replies are given when people have time.
    Sorry for the bump. My apologies

    Thank you so much for your good catch on 2b !!!!!

    as far as 2a, yes the theorem LT^{-1} \left[e^{-as} F(s)\right] = f(t - a) u(t - a) was what I was trying to apply but I just wanted to make sure I was using it correctly. Does it look like that is the case ?
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    Quote Originally Posted by DE novice View Post
    Sorry for the bump. My apologies

    Thank you so much for your good catch on 2b !!!!!

    as far as 2a, yes the theorem LT^{-1} \left[e^{-as} F(s)\right] = f(t - a) u(t - a) was what I was trying to apply but I just wanted to make sure I was using it correctly. Does it look like that is the case ?
    Not quite (to me, anyway).

    I'd present the final lines something like this:


    But LT^{-1} \left[ e^{-2s} \left( \frac{1}{2} \, \frac{1}{s} - \frac{1}{s+1} + \frac{1}{2} \, \frac{1}{(s+2)} \right) \right] = u(t-2) \, f(t - 2)

    where f(t) = LT^{-1} \left[ \frac{1}{2} \, \frac{1}{s} - \frac{1}{s+1} + \frac{1}{2} \, \frac{1}{(s+2)} \right] = \, ......

    Therefore  y = \, .......
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    Here's what I have for problem #2 listed in the original post to this thread (I have it ID'd as #3 in the attachment b/c that's what # it is in my homework). Could someone please verify my answer ? Thx.
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    Last edited by DE novice; July 23rd 2008 at 08:27 AM. Reason: added attachment
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  10. #10
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    Quote Originally Posted by DE novice View Post
    Here's what I have for problem #2 listed in the original post to this thread (I have it ID'd as #3 in the attachment b/c that's what # it is in my homework). Could someone please verify my answer ? Thx.
    Looks OK.
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