Laplace Transform from definition

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July 30th 2012, 03:43 AM
andyrobinson

Laplace Transform from definition

Hi all,

I was wondering if someone could possibly give me a step by step guide on how the laplace transfomation of f(t) = 3 + 2e^-kt is given from definition? ie using

F(s) = integral of [ f(t) . e^-st dt ]?

I can work out the actual transform but I dont follow how its done using the principal above.

Thanks !
July 30th 2012, 03:53 AM
Prove It

Re: Laplace Transform from definition

$\displaystyle \begin{align*} F(s) &= \int_0^{\infty}{e^{-s\,t}\,f(t)\,dt} \\ &= \int_0^{\infty}{e^{-s\,t}\left(3 + 2e^{-k\,t}\right)dt} \\ &= \int_0^{\infty}{3e^{-s\,t} + 2e^{-\left(s + k\right)t}\,dt} \\ &= \lim_{\epsilon \to \infty}\left[ -\frac{3}{s}e^{-s\,t} - \frac{2}{s + k}e^{-\left(s + k\right)t} \right]_0^{\epsilon} \\ &= \lim_{\epsilon \to \infty}\left[ -\frac{3}{s}e^{-\epsilon \, s} - \frac{2}{s + k}e^{-\left( s + k \right)\epsilon } \right] - \left[ -\frac{3}{s}e^{-0t} - \frac{2}{s + k}e^{-\left(s + k\right)0} \right] \\ &= 0 - \left[ -\frac{3}{s} - \frac{2}{s + k}\right] \\ &= \frac{3}{s} + \frac{2}{s + k} \end{align*}$
July 30th 2012, 08:26 AM
andyrobinson

Re: Laplace Transform from definition

thank you ! did you generate this from a website or is this your own working?
July 30th 2012, 08:44 AM
Prove It

Re: Laplace Transform from definition

Quote:

Originally Posted by andyrobinson

thank you ! did you generate this from a website or is this your own working?

I did it myself...
August 5th 2012, 03:39 PM
Phugoid

Re: Laplace Transform from definition

Quote:

Originally Posted by andyrobinson

Hi all,

I was wondering if someone could possibly give me a step by step guide on how the laplace transfomation of f(t) = 3 + 2e^-kt is given from definition? ie using

F(s) = integral of [ f(t) . e^-st dt ]?

I can work out the actual transform but I dont follow how its done using the principal above.

Thanks !

Any chance you're resitting 2nd year Mechanical Engineering maths at Glasgow Uni?