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Differential Equations with Piecewise Continuous Functions and Periodic Functions Definition 1: The unit step function u (t ) is defined by 0, t < 0 . u (t ) = 1, t ≥ 0 Suppose a ≥ 0 , what’s u (t − a ) ? 0, t < a u (t − a ) = 1, t ≥ a We still use ua (t ) to denote u (t − a ) , or ua (= t ) u (t − a ) 1, a ≤ t < b Also, we use notation uab (t ) = u (t − a ) − u (t − b) = 0, t < a, t ≥ b What’s Laplace transform of u (t − a ) ? 3, t < 2 1, 2 < t < 5 Example: Write the following function f (t ) = in terms of unit step t ,5 < t < 8 t 2 /10,8 < t functions. Property 1: Let F ( s ) = L( f )( s ) , and a ≥ 0 . We have L( f (t − a )u (t − a ))( s ) = e − as F ( s ) , and L−1 (e − as F ( s ))(t ) =f (t − a )u (t − a ) . Property 2: L( g (t )u (t − a ))( s= ) e − as L( g (t + a ))( s ) . Example 2: Determine the Laplace transform of t 2u (t − 1) . e −2 s Example : Determine L 2 . s −1 e −3 s Example : Determine L 2 . s +s−2 −1 Looking at the following differential equations coming from the modeling of on/off switches, changes. The current I in an LC series circuit is governed by the initial value problem: (1) I ''(t ) + 4 I (t )= g (t ), I (0)= 0, I '(0)= 0 where 1, 0 < t < 1, g (t ) = −1,1 < t < 2 0, 2 < +∞ Example : Solve the above initial value problem. Definition 2: A function f (t ) is said to be periodic of period of T if f (= t ) f (t + T ) for all t in the domain of f. Property 3: If f has period T and is piecewise continuous on [0,T], then ∫ L( f )( s ) = T 0 e − st f (t )dt 1 − e − sT e t , 0 < t < 1 Example : Determine L( f )( s ) , where f (t ) = and f (t ) has a period 2. 0,1 < t < 2 Example : Find the inverse Laplace transform of F (s) = 1 − e− s s (1 − e −2 s ) Example: Solve the initial value problem y ''+ π 2 y = f (t ), y (0) = 0 y '(0) = 0 1, 0 ≤ t < 1, where f (t ) = and f (t ) has period 2. 0,1 ≤ t < 2,