section%207.3

section%207.3 - Chapter 7 The laplace Transform §7.3...

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Unformatted text preview: Chapter 7. The laplace Transform §7.3 Operational Properties I 1. Translation on the s-axis: First Translation Theorem: If F (s) = L{f (t)} and a is any real number, then L{eat f (t)} = F (s − a). Ex: Evaluate. L e sin kt , Inverse Form: ￿ 5t ￿ ￿ 3t ￿ L te L−1 {F (s − a)} = eat f (t) Ex: Evaluate. 2s + 5 L−1 (s − 3)2 y ￿￿ − 6y ￿ + 9y = t2 e3t , y (0) = 2, y ￿ (0) = 17 2. Translation on the t-axis. Unit Step Function: U (t − a) = 0, 0 ≤ t < a 1, t≥a Second Translation Theorem: If F (s) = L{f (t)} and a > 0, then L{f (t − a)U (t − a)} = e−as F (s). Therefore e−as L{U (t − a)} = s Inverse Form: L−1 {e−as F (s)} = f (t − a)U (t − a) 1 Ex: Evaluate. ￿ ￿ 1 −2s −1 L e s−4 ￿ ￿ s −1 −π s/2 L e s2 + 9 Ex: Solve. y ￿ + y = f (t), where y (0) = 5 f (t) = 0, 0≤t<π 3 cos t, t≥π 2 ...
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