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N 2 n 1 test3map2302 page 2 of 4 3 15 pts a if f t

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n 2)( n 1) .
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TEST3/MAP2302 Page 2 of 4 ______________________________________________________________________ 3. (15 pts.) (a) If f ( t ) and g ( t ) are piecewise continuous functions defined for t 0, what is the definition of the convolution of f with g , ( f * g )( t )?? ( f g )( t ) t 0 f ( x ) g ( t x ) dx (b) Using only the definition of the convolution as a definite integral, not some fancy transform shenanigans, compute ( f * g )( t ) when f ( t ) = 3 t 2 and g ( t ) = 4 t . ( f g )( t ) t 0 f ( x ) g ( t x ) dx t 0 3 x 2 4( t x ) dx t 0 12 tx 2 12 x 3 dx 4 tx 3 3 x 4 t 0 t 4 . (c) Using the Laplace transform table, compute the Laplace transform of f * g when f ( t ) = t cos( t ) and g ( t ) = exp(2 t ). [Do not attempt to simplify the algebra after computing the transform.] {( f g )( t )}( s ) { f ( t )}( s ) { g ( t )}( s ) { t cos( t )}( s ) { e 2 t }( s ) s 2 1 ( s 2 1) 2 1 ( s 2) ______________________________________________________________________ 4. (10 pts.) (a) Suppose that f ( t ) is defined for t > 0. What is the definition of the Laplace transform of f , { f ( t )}, in terms of a definite integral?? { f ( t )}( s ) 0 f ( t ) e st dt lim R → ∞ R 0 f ( t ) e st dt for all s for which the integral converges. (b) Using only the definition, not the table, compute the Laplace transform of f ( t ) 0 , if 0 < t < 4 3 , if 4 < t . { f ( t )}( s ) 0 f ( t ) e st dt lim R → ∞ 4 0 0 e st dt R 4 3 e st dt lim R → ∞ 3 e 4 s s 3 e Rs s 3 e 4 s s provided s > 0. Note: You may, of course, check your "answer" using #15 in the table.
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TEST3/MAP2302 Page 3 of 4 ______________________________________________________________________ 5. (10 pts.) The equation below has a regular singular point at
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n 2 n 1 TEST3MAP2302 Page 2 of 4 3 15 pts a If f t and g t...

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