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chapter6.page02

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

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2 1. LAPLACE TRANSFORM METHODS Due the uniqueness, we can define the inverse Laplace transform L - 1 as L - 1 ( b f )( t ) = f ( t ) . Theorem 1.3 . If both b f ( s ) and b 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 ) = a b f ( s ) + b b g ( 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 0 ( t ) is piecewise continuous) for all t 0 and there are constants M, a such that | f ( t ) | ≤ Me at for t T, Then
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