7.2 notes - Chapter 7. The laplace Transform §7.2 Inverse...

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Unformatted text preview: Chapter 7. The laplace Transform §7.2 Inverse Transforms and Transforms of Derivatives 1. Inverse Transforms: Let F (s) be a Laplace transform of a function f , i.e. F (s) = L{f (t)}, we then say f (t) is the inverse Laplace transform of F (s) and write f (t) = L−1 {F (s)}. Ex: Find the Inverse Laplace transform. ￿￿ ￿￿ 1 1 1 = L−1 , t = L−1 2 s s L−1 is a linear transform: i.e. L−1 (αF (s) + β G(s)) = αf (t) + β g (t) Theorem: Some Inverse Transforms. 1. 1 = L−1 2. 3. 4. 5. ￿￿ 1 s. ￿ ￿ n! tn = L−1 sn+1 , n = 1, 2, 3, . . . . ￿ ￿ 1 eat = L−1 s−a . ￿ ￿ k sin kt = L−1 s2 +k2 , cos kt = ￿ ￿ k sinh kt = L−1 s2 −k2 , cosh kt L−1 = ￿ s s2 +k 2 . ￿ ￿ s L−1 s2 −k2 . ￿ Ex: Find the Inverse Laplace transform. ￿￿ ￿ ￿ ￿ ￿ 1 1 −2s + 6 L−1 5 , L−1 2 , L−1 s s +5 s2 + 4 ￿ ￿ s L−1 2 s + 2s − 3 2. Transforms of Derivatives Theorem: If f, f ￿ , . . . , f (n−1) are continuous on [0, ∞) and are of ex1 ponential order and if f (n) is piecewise continuous on [0, ∞), then L{f (n) } = sn F (s) − sn−1 f (0) − sn−2 f ￿ (0) − · · · − f (n−1) (0), where F (s) = L{f (t)}. 3. Solving Linear DEs: Solve DEs. 1. y ￿ + 3y = 3 sin 2t, y (0) = 2. y (0) = 1, y ￿ (0) = 5. 2. y ￿￿ − 3y ￿ + 2y = e−4t , 2 ...
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