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# 4_4a(1) - Solutions 4.4-Page 299 Problem 3 Find the...

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Solutions 4.4-Page 299 Problem 3 Find the convolution . ) ( ) ( t g t f t t g t f sin ) ( ) ( = = Eq.3 states that ( . Direct substitution into Eq.3 yields: τ τ τ d t g f t g f t ) ( ) ( ) )( 0 = = τ τ τ τ 0 )) (sin( sin ) )( ( d t t g f . To evaluate the integral, the trigonometric identity [ cos( ) cos( 2 1 sin sin B A B A = ] ) B A + can be used. Substitution of this identity into the integral yields: [ ] [ ] τ τ τ τ τ τ τ τ τ τ τ d t t d t t d t t g f t t ) cos( ) 2 cos( 2 1 ) cos( ) cos( 2 1 )) (sin( sin ) )( ( 0 0 0 = + + = = Evaluating the integral yields: [ ] [ ] ( ) ( ) [ ] ) sin( cos sin 2 1 2 1 2 1 ) cos( ) 2 cos( 2 1 2 1 2 1 0 0 cos ) 2 sin( t t t t d t t t t t t = = τ τ τ τ Simplifying yields: 2 cos sin ) )( ( t t t t g f =

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Problem 10 Apply the convolution theorem to find the inverse Laplace transforms of the functions. ) ( 1 ) ( 2 2 2 k s s s F + = Using Figure 4.1.2 on pg.268 to find the inverse Laplace functions, = = + = + = k kt t k s s k s s s F - - sin ) ( 1 1 ) ( 1 ) ( 2 2 2 1 2 2 2 1 L L τ
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4_4a(1) - Solutions 4.4-Page 299 Problem 3 Find the...

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