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Defined for t 0 what is the definition of the

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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 x 0 = 0. x 2 y xy ( x 2 1) y 0 (a) Obtain the indicial equation for the ODE at x 0 = 0 and its two roots. (b) Then use all the information available and Theorem 6.3 to say what the two nontrivial linearly independent solutions given by theorem look like without attempting to
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