4_1a - t t t f 3 ) ( + = The transforms of interest from...

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Solutions 4.1-Page 272 Problem 5 Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph). t t f sinh ) ( = The definition in (1) states that L { } = . Direct substitution yields: ) ( t f 0 ) ( dt t f e st L { } . However, sinh . Therefore the Laplace transform can be rewritten as L { } ) ( t f = 0 sinh dt t e st ) )( 2 / 1 ( t t e e t = ) ( t f = 0 ) ( 2 1 dt e e e t t st . The transform can be further simplified as follows: + = + = = + + 1 1 1 1 2 1 ) 1 ( ) 1 ( 2 1 2 1 ) ( 2 1 1 1 0 0 ) 1 ( ) 1 ( 0 s s t s t s dt e e dt e e e s e s e t s t s t t st Simplifying yields: L { } ) ( t f 1 1 2 = s

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Problem 7 Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph). The first step is to translate the graph into function form. > < = 1 0 1 0 1 ) ( t for t for t f Therefore L { } ) ( t f s e st dt e s st e s = = = 1 1 1 0 1 0 .
Problem 11 Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions. A preliminary integration by parts may be necessary.

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Unformatted text preview: t t t f 3 ) ( + = The transforms of interest from Fig. 4.1.2 are 2 1 1 ) ( ) 1 ( ) 1 ( ) ( ) ( s t s s a a s F t t f a a > + > + So the Laplace transform is 2 1 2 / 1 2 1 3 ) 1 ( s s + + + 2 ) 2 / 1 ( ) 2 / 1 ( ) 1 2 / 1 ( = = + since = ) 2 / 1 ( according to pg.267. So the Laplace transform is 2 2 / 3 3 2 s s + Problem 19 Use the transforms in Fig. 4.1.2 to find the Laplace transforms of the functions. A preliminary integration by parts may be necessary. 3 ) 1 ( ) ( t t f + = Expanding gives . The relevant transforms are: 1 3 3 ) ( 2 3 + + + = t t t t f 1 2 ! 1 1 1 ) ( ) ( + n n s n t s t s s F t f The transform is s s s s s F 1 3 ) ! 2 ( 3 ! 3 ) ( 2 3 4 + + + = s s s s s F 1 3 6 6 ) ( 2 3 4 + + + =...
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This note was uploaded on 06/06/2011 for the course EGM 3311 taught by Professor Haftka during the Spring '11 term at University of Florida.

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4_1a - t t t f 3 ) ( + = The transforms of interest from...

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