Sol_HW3

# Sol_HW3 - 3.2 Find the Laplace transform of the following...

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1 2 2 23 3.2 Find the Laplace transform of the following time functions: Solution: ( ) ( ) 3 7 ( ) { ( )} {3} {7 } { } { ( )} 372 ! 1 bf t tt t ft t t t ss s δ =+ + + =+ + + 33 3 3 3 32 3 37 2 sss s ++ + = 22 3.3 Find the Laplace transform of the following time functions: Solution: ( ) ( ) sin 3 { ( )} { } { sin3 } 2! 3 (2 )9 t t cf t t e t t e t ss 3 3.4 Find the Laplace transform of the following time functions: Solution: ( ) ( ) 2 cos Use the following Laplace transforms and properties (Table A.1, entries 4,11, and 3), t t t e 2 2 1 { } () { ( )} ( ) {cos } { ( )} { } 2 { cos } 1 2( (1 ) at t te sa d tg t G s ds s at t e t t d sd = + =− = + + 3 3 3 3 2 2 2 ) 1 11 2 ) ( 1 ) s s + 0 0 3.5 Find the Laplace transform of the following time functions (* denotes convolution): Solution: ( ) ( ) cos( )sin { ( )} { cos( } {cos *sin } t t ef t t d t d t t ττ τ = 3 42 This is just the deTnition of the convolution theorem, 1 { ( )} 2 1 == 3

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2 3.6 Given that the Laplace transform of ( ) is ( ), find the Laplace transform of the following: Solution: ( ) ( ) ( )cos First write cos in terms of the related Euler identi ft Fs ag t f t t t = ty (Eq. B.33), 11 ( ) ( )cos ( ) ( ) ( ) 22 2 Then using entry 4 of Table A.1 we have, 111 ( ) ( ) ( ) ( ) ( 222 jt jt jt
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Sol_HW3 - 3.2 Find the Laplace transform of the following...

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