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

Question

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

unklerhaukus · Answer

(b) partial fractions
$$\begin{align*}
F(p)&=\frac{p+2}{2p^2+p-1}\
&=\frac{p+2}{(p+1)(2p-1)}&=\frac{A}{p+1}+\frac{B}{2p-1}\
\&&p+2=A(2p-1)+B(p+1)\
\&\text{for } p=-1&1=-3A\
&&A=-\tfrac{1}3\
\&\text{for } p=0&2=-A+B\
&&B=2-\tfrac 13=\tfrac53\
\
F(p)&=\frac{-\tfrac13}{p+1}+\frac{\tfrac53}{2p-1}\
&=\tfrac13\left(\frac{5}{2p-1}-\frac{1}{p+1}\right)\
\
f(t)&=\tfrac13\mathcal L^{-1}\left\{\frac{5}{2p-1}-\frac{1}{p+1}\right\}\
&=\tfrac13\left(\tfrac52\mathcal L^{-1}\left\{\frac{1}{p-\tfrac12}\right\}-\mathcal L^{-1}\left\{\frac1{p+1}\right\}\right)\
&=\tfrac13\left(\tfrac52e^{t/2}-e^{-t}\right)\
\
&=\frac56e^{t/2}-\frac13e^{-t}\
\end{align*}
$$

anonymous · Answer

http://www.wolframalpha.com/input/?i=inverse+laplace+transform+of+%28s%2B2%29%2F%282s%5E2%2Bs-1%29

unklerhaukus · Answer

what do i do with that/??

unklerhaukus · Answer

im having trouble with deconvolution

anonymous · Answer

|dw:1353549631736:dw|