chapter6.page02

chapter6.page02 - 2 1 LAPLACE TRANSFORM METHODS Due the...

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Unformatted text preview: 2 1. LAPLACE TRANSFORM METHODS Due the uniqueness, we can define the inverse Laplace transform L-1 as L-1 (f )(t) = f (t). Theorem 1.3. If both f (s) and g(s) exist for all s > c, then af (t)+ bg(t) has Laplace transform for all constant a and b and af + bg(s) = af (s) + bg(s), for all s > c So to find Laplace transform of summation, we just need to find Laplace transform of each term. The next result shows that Laplace transform changes derivative into scalar multiplication, it is this property enable Laplace transform to change ODE into algebraic equation. Theorem 1.4. Suppose that the function f (t) is continuous and piecewise smooth (f (t) is piecewise continuous) for all t 0 and there are constants M, a such that |f (t)| M eat for t T, Then f (s) is defined for all s > a and f (s) = sf (s) - f (0). Now if nth derivative f n (t) is piecewise smooth, then f (n) (s) = sn f (s) - sn-1 f (0) - sn-2 f (0) - - f (n-1) (0). For example, When n = 2 f (2) (s) = s2 f (s) - sf (0) - f (0). When n = 3 f (3) (s) = s3 f (s) - s2 f (0) - sf (0) - f (0). When n = 4 f (4) (s) = s4 f (s) - s3 f (0) - s2 f (0) - sf (0) - f (3) (0). The following table list most commonly used functions, and ua (t) = u(t - a) = is the Heaviside function. 0 if 1 if t<a ta ...
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