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Laplace Transform equation help

  • #1
198
0
[SOLVED] Laplace Transform

Homework Statement


Suppose F(s) = [tex]\displaystyle\mathcal{L}(f(t)) [/tex]
Show that [tex]\displaystyle\mathcal{L}(f(ct)) = 1/c F(s/c) [/tex]


Homework Equations





The Attempt at a Solution



[tex]\displaystyle\mathcal{L}(f(t)) = \int_0^{inf} e^{-st} f(t) dt[/tex]
[tex]\displaystyle\mathcal{L}(f(ct)) = \int_0^{inf} e^{-st} f(ct) dt[/tex]
I'm not quite sure what to do after this...

I could play around with integration by parts, but in this case I don't think it yields anything useful
[tex]\frac{F(ct)}{c} e^{-st} - \int \frac{F(ct)}{c} (-s e^{-st} ) dt[/tex]

[tex]{F(t)e^{-st} + \int F(t) se^{-st} dt}{}[/tex]
 
Last edited:

Answers and Replies

  • #2
761
13
Suppose [tex]u=ct [/tex] then we get:

[tex] \int_0^{ \infty} e^{-st} f(ct)\ \mbox{d}t = \frac{1}{c} \int_0^{\infty}\ e^{\frac{-s}{c} u}\ f(u)\ \mbox{d}u = \frac{1}{c}\ F \left( \frac{s}{c} \right) [/tex]
 

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