undefined Solution The Laplace transform of the function f ( t ) = 5 cosh 2 t − 3 sinh t . The Laplace transform of the function f ( t ) = 5 cosh 2 t − 3 sinh t is 5 [ s s 2 − 4 ] − 3 [ 1 s 2 − 1 ] _ . Definition used: The Laplace transform of the function f ( t ) is defined by the function F ( s ) and is denoted by L ( f ) and is given by the formula F ( s ) = L ( f ) = ∫ 0 ∞ e − s t f ( t ) Theorem used: Linearity of Laplace Transform: Let f ( t ) and g ( t ) be two functions whose transforms exist and a , b any constants then the transform of a f ( t ) + b g ( t ) exists and is given by the formula L ( a f ( t ) + b g ( t ) ) = a L ( f ( t ) ) + b L ( g ( t ) ) Formula used: The functions and their Laplace transforms are given as follows f ( t ) L ( f ) cosh a t s s 2 − a 2 sinh a t a s 2 − a 2 Calculation: The given function is f ( t ) = 5 cosh 2 t − 3 sinh t . Let f ( t ) = 5 f 1 ( t ) − 3 f 2 ( t ) , where f 1 ( t ) = cosh 2 t and f 2 ( t ) = sinh t . Note that f 1 ( t ) = cosh 2 t is of the form cosh a t , where a = 2 . Substitute a = 2 in the above formula to obtain the Laplace transform of the function f 1 ( t ) = cosh 2 t as follows: L ( f 1 ) = L ( cosh 2 t ) = s s 2 − 2 2 = s s 2 − 4 Note that f 2 ( t ) = sinh t is of the form sinh a t , where a = 1 . Substitute a = 1 in the above formula to obtain the Laplace transform of the function f 2 ( t ) = sinh t as follows: L ( f 2 ) = L ( sinh t ) = 1 s 2 − 1 2 = 1 s 2 − 1 Use the linearity of Laplace transform to obtain the Laplace transform of the function f ( t ) = 5 cosh 2 t − 3 sinh t as follows: L ( f ) = L ( 5 cosh 2 t − 3 sinh t ) = L ( 5 cosh 2 t ) − L ( 3 sinh t ) = 5 L ( cosh 2 t ) − 3 L ( sinh t ) = 5 ( s s 2 − 4 ) − 3 ( 1 s 2 − 1 2 ) Therefore, the Laplace transform of the function f ( t ) = 5 cosh 2 t − 3 sinh t is 5 [ s s 2 − 4 ] − 3 [ 1 s 2 − 1 ] _ .

10th Edition

ISBN: 9781119455929

Author: Kreyszig

Publisher: WILEY

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