050307-132a-clab 4

050307-132a-clab 4 - Solve the following set of equations...

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C HEMICAL E NGINEERING 132A Professor Todd Squires Spring Quarter, 2007 Computer Lab Assignment 4 Laplace Transforms P RACTICE WITH L APLACE TRANSFORMS New Commands: FullSimplify[%] A = {{1,2},{2,3}} (*two-by-two matrix*) MatrixForm[A] v = {4,5} (*Column Vector*) A.v (*Matrix Multiplication*) Inverse[A] (*Matrix Inversion*) Compute the following Laplace Transforms by hand, then use Mathematica to verify. First shifting theorem : compute the Laplace transform of a . e - 2 t t 2 (1) b . e - t sin t (2) c . e - 5 t (3) Compute the inverse transform of a . 1 s 2 +2 s +1 (4) b . 1 1+( s - 2) 2 (5) Second Shifting Theorem: Compute the Laplace Transform of a . H ( t - 5) cos( t - 5) (6) b . H ( t - 2) t (7) Compute the inverse Laplace Transform of a . e - s s ( s - 1) (8) b . e - s s +1 (9) Convolutions: Compute the Inverse Laplace Transform of F ( s ) G ( s ) directly, then by using the convolution theorem: 1) F ( s ) = 1 /s 2 , G ( s ) = 1 / ( s + 1) (10) 2) F ( s ) = 1 / ( s + 1) , G ( s ) = 1 / ( s + 2) (11) (12) Extra credit bonus problem: Systems of equations:
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Unformatted text preview: Solve the following set of equations dx dt + 2 dy dt-3 dz dt = 2 (13) dy dt-3 dz dt + 2 x = 0 (14) dx dt + dz dt + 3 z = 3 (15) x (0) = 0 , y (0) = 0 , z (0) = 0 (16) by 1) Laplace-transforming each equation by hand 2) Writing as a matrix equation A . x = b , where A is a three-by-three matrix, x = { X ( s ) , Y ( s ) , Z ( s ) } is a column vector, and b is a one-by-three column vector. Enter A and b into Mathematica. 3) Solve the equation by inverting the matrix: x = A-1 . b . That is, compute the inverse of A using mathematica, multply it by b . This gives the laplace-transformed solutions { X ( s ) , Y ( s ) , Z ( s ) } . 4) Invert the solutions to give the solutions { x ( t ) , y ( t ) , z ( t ) } ....
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