3 - laplace_disc_steps

# 3 - laplace_disc_steps - Step functions Recall the...

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Step functions April 9, 2011 Recall the Heaviside step function (or unit-step function) is deﬁned as u c ( t ) = ( 0 x < c 1 x > c (1) Also recall the “t-shifting” theorem (as opposed to the “s-shifting” theorem): L { u c ( t ) f ( t - c ) } = e - cs L { f ( t ) } . (2) In order to solve diﬀerential equations with discontinuous forcing functions, we express them in terms of the Heaviside functions, and use above theorem to take their Laplace transforms. Problem statement: (“Forward” problem) Find the Laplace transform of g ( t ) = u c ( t ) h ( t ) (3) Steps: 1. Identify f ( t - c ) . This function is what happens to be multiplied by u c ( t ) . In problem statement above, f ( t - c ) = h ( t ) . 2. Evaluate f ( t ) from Step 1. This means replacing t with t + c in the identiﬁed function. 3. From Eq. ( 2 ), L { g ( t ) } = e - cs L { f ( t ) } , where f ( t ) is the func- tion determined in Step 2. Problem statement:

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3 - laplace_disc_steps - Step functions Recall the...

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