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Laplace transfoms
How do you solve
y'' + 4y' + y + 4(integral from 0 to t of sin(t-s)y''(s) ds) = cost
y(0)=y'(0)=0
' denotes differentiation wrt t
y'' + 4y' + y + 4(integral from 0 to t of sin(t-s)y''(s) ds) = cost
y(0)=y'(0)=0
' denotes differentiation wrt t
Let F(s) be the Laplace transform of y(t).
Hints to get you started
(i) By the convolution theorem, the Laplace transform of (integral from 0 to t) sin(t - s)y''(s) ds is s^2 F(s)/(s^2 + 1). (ii) The Laplace transform of t^n e^(kt) is n!/(s - k)^(n + 1). (iii) All the standard results you need are at http://www.engineering.com/content/ContentDisplay?contentId=41005009.
Solution
s^2 F(s) + 4s F(s) + F(s) + 4(1/(s^2 + 1))(s^2 F(s)) = s/(s^2 + 1)
F(s) (s^2 + 4s + 1 + 4s^2/(s^2 + 1)) = s/(s^2 + 1)
F(s) ((s^2 + 4s + 1)(s^2 + 1) + 4s^2) = s
F(s)
= s/((s^2 + 4s + 1)(s^2 + 1) + 4s^2)
= s/(s + 1)^4
= 1/(s + 1)^3 - 1/(s + 1)^4
y(t)
= (1/2)t^2 e^(-t) - (1/6)t^3 e^(-t)
Hints to get you started
(i) By the convolution theorem, the Laplace transform of (integral from 0 to t) sin(t - s)y''(s) ds is s^2 F(s)/(s^2 + 1). (ii) The Laplace transform of t^n e^(kt) is n!/(s - k)^(n + 1). (iii) All the standard results you need are at http://www.engineering.com/content/ContentDisplay?contentId=41005009.
Solution
s^2 F(s) + 4s F(s) + F(s) + 4(1/(s^2 + 1))(s^2 F(s)) = s/(s^2 + 1)
F(s) (s^2 + 4s + 1 + 4s^2/(s^2 + 1)) = s/(s^2 + 1)
F(s) ((s^2 + 4s + 1)(s^2 + 1) + 4s^2) = s
F(s)
= s/((s^2 + 4s + 1)(s^2 + 1) + 4s^2)
= s/(s + 1)^4
= 1/(s + 1)^3 - 1/(s + 1)^4
y(t)
= (1/2)t^2 e^(-t) - (1/6)t^3 e^(-t)
Original post by Zacklen
let e^x = z then ur done
is this a joke?
Original post by TeeEm
is this a joke?
nah thats how you do integration by parts
Original post by TeeEm
is this a joke?
Would you like to call the asylum, or shall I?
Original post by 16Characters....
Would you like to call the asylum, or shall I?
He's been carted off, cleaning up now. Some people really have nothing better to do.
Original post by 16Characters....
Would you like to call the asylum, or shall I?
Already banned as he was a troll ...
(he got me at first, then I saw the date ...)
Original post by Zacken
He's been carted off, cleaning up now. Some people really have nothing better to do.
Original post by TeeEm
Already banned as he was a troll ...
(he got me at first, then I saw the date ...)
(he got me at first, then I saw the date ...)
Trolling a maths forum. Bet the girls love his exciting lifestyle.
Original post by 16Characters....
Trolling a maths forum. Bet the girls love his exciting lifestyle.
truly sad ...
(I mean it)
as this thread is not likely to be removed now I might as well put some links for anyone who might be looking for Laplace transforms
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_introduction.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_further.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_applications.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_introduction.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_further.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_applications.pdf
Original post by TeeEm
as this thread is not likely to be removed now I might as well put some links for anyone who might be looking for Laplace transforms
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_introduction.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_further.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_applications.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_introduction.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_further.pdf
http://www.madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms_applications.pdf
Lovely. PRSOM. (the thread you tagged me on is being taken care of).
Original post by Zacken
Lovely. PRSOM. (the thread you tagged me on is being taken care of).
no worries
Original post by TeeEm
is this a joke?
This thread is from 2006!
Original post by spotify95
This thread is from 2006!
There was a poster earlier who was replying to loads of old threads. This was one of them.
He's banned now,
Original post by spotify95
This thread is from 2006!
Original post by notnek
There was a poster earlier who was replying to loads of old threads. This was one of them.
He's banned now,
He's banned now,
Okay, so it looks like they removed his post too.
In which case apologies @TeeEm for thinking it was you!
Original post by spotify95
Okay, so it looks like they removed his post too.
In which case apologies @TeeEm for thinking it was you!
In which case apologies @TeeEm for thinking it was you!
Original post by spotify95
Okay, so it looks like they removed his post too.In which case apologies @TeeEm for thinking it was you!
Original post by TeeEm
see ...and since the thread was forgotten and was not blocked I posted 3 links on Laplace transforms if future people do a search on it
Which is great, so thanks for the help
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