# Laplace Transform equation help

[SOLVED] Laplace Transform

## Homework Statement

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

## The Attempt at a Solution

$$\displaystyle\mathcal{L}(f(t)) = \int_0^{inf} e^{-st} f(t) dt$$
$$\displaystyle\mathcal{L}(f(ct)) = \int_0^{inf} e^{-st} f(ct) dt$$
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
$$\frac{F(ct)}{c} e^{-st} - \int \frac{F(ct)}{c} (-s e^{-st} ) dt$$

$${F(t)e^{-st} + \int F(t) se^{-st} dt}{}$$

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Suppose $$u=ct$$ then we get:
$$\int_0^{ \infty} e^{-st} f(ct)\ \mbox{d}t = \frac{1}{c} \int_0^{\infty}\ e^{\frac{-s}{c} u}\ f(u)\ \mbox{d}u = \frac{1}{c}\ F \left( \frac{s}{c} \right)$$