PropLaplaceTransform

# PropLaplaceTransform - Properties of the Laplace Transform...

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Properties of the Laplace Transform 1) The Linear Property Let f(t) and g(t) be functions whose Laplace transforms exit for s > a 1 and s > a 2 respectively. Then, for s > max{ a 1 , a 2 } and c 1 and c 2 any constants L {c 1 f(t) + c 2 g(t)} = e ! st {c 1 f(t) + c 2 g(t)}dt 0 " # = c 1 e ! st f(t)dt 0 " # + c 2 e ! st g(t)dt = 0 " # c 1 L {f(t)} + c 2 L {g(t)} This means the the Laplace transform is a linear operator. Example : 1) L {5e -2t - 3 sin 4t} = 5 L {e-2t} - 3 L {sin 4t} = 5 s + 2 ! 12 s 2 + 16 , s > 0. 2) L {cos 2 x} = L {1/2(1 + cos 2x)} = ½ L {1} + 1/2 L {cos 2x} = 1 2s + 1 2 s s 2 + 4 = 2s 2 + 4 2s(s 2 + 4) = s 2 + 2 s(s 2 + 4) . 2) Laplace Transform of the Derivatives of f . Theorem : Suppose f(t) is continuous and its derivative f’ is piecewise continuous on any interval 0 t A. Suppose further that there exist constants M, a, and t 0 such that |f(t)| Me -at for t > t 0 . Conclusion : Then, L {f’(t)} exists for s > a, and moreover L {f’(t)} = s L {f(t)} – f(0) Proof : If f’(t) has points of discontinuity in the interval 0 t A, let them be denoted by t 1 , t 2 , … , t n . Then, we can write the integral e ! st " f (t)dt = e ! st "

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