notes 6-3

notes 6-3 - EXAMPLE Sketch the graph of the function f t-π...

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SECTION 6.3 STEP FUNCTIONS AND TRANSLATIONS Because of the applications of the differential equation y 00 + ay 0 + by = g ( t ) in mechanical vibration and electrical circuit problems, the function g ( t ) is sometimes called the forcing function. We need to learn how to handle forcing functions that arise when magnets or voltages are switched on and off, and to do so we need to look at the most basic discontinuous function together with translations of functions. The unit step function or Heaviside function is denoted by u c for any c 0 and is defined by u c ( t ) = ( 0 t < c 1 t c Its graph looks like: EXAMPLE. Suppose we turn on a forcing function sin( t - 1) at t = 1. What does the graph of this forcing function look like, and how can we give a formula for it?

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Unformatted text preview: EXAMPLE. Sketch the graph of the function f ( t-π ) u π ( t ), where f ( t ) = t 2 . TABLE 6.2.1, #13. L{ u c ( t ) f ( t-c ) } = e-cs L{ f ( t ) } = e-cs F ( s ) TABLE 6.2.1, #14. If F ( s ) = L{ f ( t ) } and if c is a constant, then L{ e ct f ( t ) } = F ( s-c ). EXAMPLE. Find the Laplace transform of the function f ( t ) = , t < π t-π, π ≤ t < 2 π , t ≥ 2 π . EXAMPLE. Find the inverse Laplace transform of the function F ( s ) = ( s-2) e-3 s s 2-4 s + 3 . HOMEWORK: SECTION 6.3...
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