Question

Find the function f(x) which has the following Laplace transform:
F (s) = (4s+3)/(s^2-2s+8)

Answer 1

lol this is bringing back my differential equations days...i'll try

Answer 2

You have to take the inverse laplace transform of F(s)

Answer 3

haha. Personally I find it a hateful section! :) Don't you have to break up the fraction in some way. If the 8 on the bottom was negative it would be so much easir!! :)

Answer 4

(partial fraction) but i cant seem to do it...

Answer 5

Yeah the bottom function is not factorable so we'll need to complete that expression as a complete square

Answer 6

Okay when you complete the square in the denominator you get:
s^2 - 2x + 8 = (s - 1)^2 + 7

Answer 7

I'm sure that the laplace will transform into either sin or cos so we're going to work on the top as well

Answer 8

[4(s - 1) + 4 + 3 ] / (s - 1)^2 + 7

Answer 9

4(s - 1) / (s - 1)^2 +7

Remember that 7 = [sqrt(7)]^2

Answer 10

Also we have 7 / [(s-1)^2 + 7]

Answer 11

For 4(s-1) / (s -1)^2 + 7 the inverse laplace transform is
\[4e ^{t}\cos (\sqrt{7}t)\]

Answer 12

For the second one: 7 = sqrt[7] x sqrt [7]

Answer 13

So we can write it as \[\sqrt{7}[\sqrt{7}/[(s - 1)^2 + (\sqrt{7})^{2}]\]

Answer 14

So that leaves us with \[\sqrt{7}e ^{t}\sin(\sqrt{7}t)\]

Answer 15

So your overall answer is f(t) = 4e^tcos(7√t) + √7e^tsin(7√t)

Do you get it?

Answer 16

YES!!! I DID!!!! I can't thank you enough!!! Wish I could give you more than one medal!!!! :)

Answer 17

No problem! It's good you understand...I thought I will have to attach a file because explanation seemed long