UnkleRhaukus 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) $\frac{p+2}{2p^2+p-1}$

1. UnkleRhaukus

(b) partial fractions \begin{align*} F(p)&=\frac{p+2}{2p^2+p-1}\\ &=\frac{p+2}{(p+1)(2p-1)}&=\frac{A}{p+1}+\frac{B}{2p-1}\\ \\&&p+2=A(2p-1)+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}{2p-1}\\ &=\tfrac13\left(\frac{5}{2p-1}-\frac{1}{p+1}\right)\\ \\ f(t)&=\tfrac13\mathcal L^{-1}\left\{\frac{5}{2p-1}-\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*}

2. malevolence19
3. UnkleRhaukus

what do i do with that/??

4. UnkleRhaukus

im having trouble with deconvolution

5. mahmit2012

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6. mahmit2012

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7. mahmit2012

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8. mahmit2012

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9. mahmit2012

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10. UnkleRhaukus

(b) table of laplace transforms \begin{align*} F(p)&=\frac{p+2}{2p^2+p-1}\\ &=\frac{p+2}{(2p-1)(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*}

11. UnkleRhaukus

the derivative fo the dirac delta is not familiar to me

12. malevolence19

Yeah, I was wondering the same thing. How is $\delta ' (t)$ defined?