Question

Find the Laplace Transform F(s) of:
f(x)=6sin3x+4cos5x-2e^-7x+8xe^2x+7x^3

Answer 1

you're not expected to find them using the definition (integrating to infinity)
you can use a table of transforms:

\[F(s) = 6*\frac{ 3 }{ s^2 + 3^2 } + 5*\frac{ s^2 }{ s^2 +5^2 } + \frac{ 2 }{ s + 7 } + \frac{ 8 }{ (s -2)^2 } + \frac{ 7 * 3! }{ s^4 }\]

Answer 2

um can you explain what you did there?

Answer 3

$$\mathcal{L}\{6\sin3x+4\cos5x-2e^{-7x}+8xe^{2x}+7x^3\}$$Take it term by term:$$\mathcal{L}\{6\sin3x\}=6\cdot\mathcal{L}\{\sin3x\}=\frac{18}{s^2+9}\\\mathcal{L}\{4\cos5x\}=\frac{5s}{s^2+25}\\\mathcal{L}\{-2e^{-7x}\}=-\frac2{s+7}\\\mathcal{L}\{8xe^{2x}\}=\frac8{(s-2)^2}\\\mathcal{L}\{7x^3\}=\frac{7\cdot3!}{s^4}=\frac{42}{s^4}$$... putting it all together we have:$$F(s)=\frac{18}s^2+\frac{5s}{s^2+25}-\frac2{s+7}+\frac8{(s-2)^2}+\frac{42}{s^4}$$Here's a useful table: http://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms

Answer 4

oops the first term came out broken! $$F(s)=\frac{18}{s^2+9}+\dots$$

Answer 5

simply used the table which defines what the transformation is.

you can also use the definition; but the results of all continuous functions are well known.

i personally used this table right now; i use the one in my textbook for homework: https://controls.engin.umich.edu/wiki/images/9/97/Table_5_3_Single_sided_Laplace_Transforms.jpg

Answer 6

i made a mistake in mine too. the second term is s/(s^2 + 5^2) not s^2. sorry

Answer 7

My second term should be \(\dfrac{4s}{s^2+25}\)... woops.

Answer 8

hmm am trying to understand it, but thanks oldrin

Answer 9

No problem. Since the Laplace transform is a linear transformation (like differentiation and integration), you can take it term by term and constant factors pass through.

Answer 10

ok but may i know what is the "s" is? am kind of a beginner here

Answer 11

oops sorry for the lack of response. \(s\) is a "complex frequency" -- I wouldn't worry about it to much. The idea is that our Laplace transform puts us in a new space, the \(s\)-domain, in which differential equations reduce to algebraic ones.