undefined Solution To graph: The function sin t ( π 2 < t < π ) ; represent it using unit step function and to find the transform of the given function. The given function represented in terms of unit step function is sin t ( u ( t − π 2 ) − u ( t − π ) ) _ and the Laplace transform of the given function is s e − π 2 s s 2 + 1 + e − π s s 2 + 1 _ . Formula used: 1. The Laplace transform of the function 1 s n is 1 s n = t n − 1 ( n − 1 ) ! . Definition used: Unit step function or Heaviside Function: The unit step function or the Heaviside function denoted by u ( t − a ) is defined as follows. u ( t − a ) = { 0 , if t < a 1 , if t > a } Second shifting theorem If f ( t ) has the transform F ( s ) , then the shifted function f ˜ ( t ) = f ( t − a ) u ( t − a ) = { 0 if t < a f ( t − a ) if t > a } has the transform e − a s F ( s ) . That is, if L ( f ( t ) ) = F ( s ) then, L ( f ( t − a ) u ( t − a ) ) = e − a s F ( s ) ⋯ ( 1 ) Calculation: The given function is sin t ( π 2 < t < π ) . Use the above definition of unit step function and rewrite the given function in terms of unit step function as shown below. sin t ( u ( t − π 2 ) − u ( t − π ) ) = sin ( ( t + π 2 ) − π 2 ) u ( t − π 2 ) − sin ( t + π − π ) u ( t − π ) Take Laplace transform on the right hand side of the above equation; apply the second shifting theorem; use the formula used and obtain the Laplace transform of the given function as shown below. . L ( sin t ) = L ( sin ( ( t + π 2 ) − π 2 ) u ( t − π 2 ) − sin ( t + π − π ) u ( t − π ) ) = e − π 2 s L ( sin ( t + π 2 ) ) − e − π s L ( sin ( t + π ) ) = e − π 2 s L ( cos t ) − e − π s L ( − sin t ) ( ∵ sin ( t + π 2 ) = cos t sin ( t + π ) = − sin t ) = s e − π 2 s s 2 + 1 + e − π s s 2 + 1 Thus, the given function represented in terms of unit step function is sin t ( u ( t − π 2 ) − u ( t − π ) ) _ and the Laplace transform of the given function is s e − π 2 s s 2 + 1 + e − π s s 2 + 1 _ . Now, use the online graphing calculator and sketch the given function as shown below. Thus, the Figure 1 is the required graph of the function sin t ( π 2 < t < π ) .

10th Edition

ISBN: 9781119664697

Author: Kreyszig

Publisher: WILEY

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