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Thread: Can I get a good explication of this Laplace transform?

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    Can I get a good explication of this Laplace transform?

    Can I get a good explication of this Laplace transform?-ex.png
    I found it at the end of this website Differential Equations - Laplace Transforms
    I felt good about the other practice problems on the site but this one is over my head. Maybe it can be written with more in depth steps? Thanks.
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    Re: Can I get a good explication of this Laplace transform?

    Attachment 38202

    Sorry.. Dont know what going on with these multiple uploads of same picture. So is this reasoning sound? specifically steps 2 to 3 because I think that was what was most confusing. Also, the equation says that you need f(t) but we are using f'(t). Is this still ok?
    Attached Thumbnails Attached Thumbnails Can I get a good explication of this Laplace transform?-20171015_101050.jpg  
    Last edited by topsquark; Oct 15th 2017 at 10:14 AM. Reason: What multiple uploads? :)
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    Re: Can I get a good explication of this Laplace transform?

    I don't know how you got all three of them there but if you need to remove some then scroll down the page for a bit until you see the button "manage attachments." Click on the image you want removed and you'll see a little x in the top right. Click that and the image is removed. Remember to save the post afterward.

    -Dan
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    Re: Can I get a good explication of this Laplace transform?

    The Laplace transform of f(x) is defined as $L(f)= \int_0^\infty f(t)e^{-st} dt$. To find the Laplace transform of f'(t), we can use "integration by parts". $L(f')= \int_0^\infty f'(t)e^{st}dt$. Use integration by parts with $u(t)= e^{-st}$ and $dv(t)= f'(t)dt$. Then $du= -se^{-st}$ and $v= f(t)$ so $\int_0^\infty f'(t)e^{-st}dt= \left[e^{-st}f(t)\right]_0^\infty+ s\int_0^\infty f(t)e^{-st}dt= f(0)+ sL(f)$.
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