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 2 years ago
Use partial fractions or the convolution theorem and the table of Laplace transforms, to find functions of \(t\) which have the following Laplace transforms.
(b) \[\frac{p+2}{2p^2+p1}\]
 2 years ago
Use partial fractions or the convolution theorem and the table of Laplace transforms, to find functions of \(t\) which have the following Laplace transforms. (b) \[\frac{p+2}{2p^2+p1}\]

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UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.0(b) partial fractions \[\begin{align*} F(p)&=\frac{p+2}{2p^2+p1}\\ &=\frac{p+2}{(p+1)(2p1)}&=\frac{A}{p+1}+\frac{B}{2p1}\\ \\&&p+2=A(2p1)+B(p+1)\\ \\&\text{for } p=1&1=3A\\ &&A=\tfrac{1}3\\ \\&\text{for } p=0&2=A+B\\ &&B=2\tfrac 13=\tfrac53\\ \\ F(p)&=\frac{\tfrac13}{p+1}+\frac{\tfrac53}{2p1}\\ &=\tfrac13\left(\frac{5}{2p1}\frac{1}{p+1}\right)\\ \\ f(t)&=\tfrac13\mathcal L^{1}\left\{\frac{5}{2p1}\frac{1}{p+1}\right\}\\ &=\tfrac13\left(\tfrac52\mathcal L^{1}\left\{\frac{1}{p\tfrac12}\right\}\mathcal L^{1}\left\{\frac1{p+1}\right\}\right)\\ &=\tfrac13\left(\tfrac52e^{t/2}e^{t}\right)\\ \\ &=\frac56e^{t/2}\frac13e^{t}\\ \end{align*} \]

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.0http://www.wolframalpha.com/input/?i=inverse+laplace+transform+of+%28s%2B2%29%2F%282s%5E2%2Bs1%29

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.0what do i do with that/??

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.0im having trouble with deconvolution

mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1353549631736:dw

mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1353549766387:dw

mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1353549812275:dw

mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1353550099053:dw

mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1353550140216:dw

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.0(b) table of laplace transforms \[\begin{align*} F(p)&=\frac{p+2}{2p^2+p1}\\ &=\frac{p+2}{(2p1)(p+1)}\\ &=\tfrac12\times\frac{p}{(p\tfrac12)(p+1)}+\frac{1}{(p\tfrac12)(p+1)}\\ f(t)&=\tfrac12\mathcal L^{1}\left\{\frac{p}{(p\tfrac12)(p+1)}\right\}+\mathcal L^{1}\left\{\frac{1}{(p\tfrac12)(p+1)}\right\}\\ &=\tfrac12\times\frac{1/2e^{t/2}+e^{t}}{1/2+1}+\frac{e^{t/2}e^{t}}{1/2+1}\\ &=\tfrac12\times\frac{1/2e^{t/2}+e^{t}}{3/2}+\frac{e^{t/2}e^{t}}{3/2}\\ &=\tfrac13\times\left(1/2e^{t/2}+e^{t}\right)+\tfrac23{e^{t/2}\tfrac23e^{t}}\\ &=\tfrac16e^{t/2}+\tfrac13e^{t}+\tfrac23{e^{t/2}\tfrac23e^{t}}\\ &=\frac56e^{t/2}\frac13e^{t} \end{align*}\]

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.0the derivative fo the dirac delta is not familiar to me

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah, I was wondering the same thing. How is \[\delta ' (t)\] defined?
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