undefined Solution The Laplace transform of the function f ( t ) = sin 2 ( 1 2 π t ) . The Laplace transform of the function f ( t ) = sin 2 ( 1 2 π t ) is 1 2 ( 1 s − s s 2 + π 2 ) _ . Definition used: The Laplace transform of the function f ( t ) is defined by the function F ( s ) and is denoted by L ( f ) and is given by the formula F ( s ) = L ( f ) = ∫ 0 ∞ e − s t f ( t ) Theorem used: Linearity of Laplace Transform: Let f ( t ) and g ( t ) be two functions whose transforms exist and a , b any constants then the transform of a f ( t ) + b g ( t ) exists and is given by the formula L ( a f ( t ) + b g ( t ) ) = a L ( f ( t ) ) + b L ( g ( t ) ) Formula used: The functions and their Laplace transforms are given as follows f ( t ) L ( f ) cos ω t s s 2 + ω 2 1 1 s Calculation: The given function is f ( t ) = sin 2 ( 1 2 π t ) . Simplify further as, f ( t ) = sin 2 ( 1 2 π t ) = 1 2 ( 1 − cos 2 ( 1 2 π t ) ) = 1 2 ( 1 − cos π t ) Let f 1 ( t ) = cos π t , which is of the form cos ω t , where ω = π . Substitute ω = π in the above formula to obtain the Laplace transform of the function f 1 ( t ) = cos π t as follows: L ( f 1 ) = L ( cos π t ) = s s 2 + π 2 Use the linearity of Laplace transform to obtain the transform of the function f ( t ) = 1 2 ( 1 − cos π t ) as follows: L ( f ) = L ( 1 2 ( 1 − cos π t ) ) = 1 2 ( L ( 1 ) − L ( cos π t ) ) = 1 2 ( 1 s − s s 2 + π 2 ) Therefore, the transform of the function f ( t ) = sin 2 ( 1 2 π t ) is 1 2 ( 1 s − s s 2 + π 2 ) _

Custom Kreyszig: Advanced Engineering Mathematics

10th Edition

ISBN: 9781119166856

Author: Kreyszig

Publisher: JOHN WILEY+SONS INC.CUSTOM

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Page of 2 ZOOM + 1) a) Answer the following questions by circling TRUE or FALSE (No explanation or work required). [1 0 0 i) A = 0 2 6 is invertible. (TRUE FALSE) LO -4-12] ii) We can use the transpose of the cofactor matrix to find the inverse of a matrix. (TRUE FALSE) = iii) If A 2, and A is a 5x5 square matrix, |2A] = 64. (TRUE FALSE) iv) Every vector space must contain two trivial subspaces. (TRUE FALSE) v) The set of all integers with standard operations is a vector space. (TRUE FALSE) b) Write v as a linear combination of the vectors in the set S, if possible, where v=(1,-4), and S={(1,2),(1,-1)}. 2) a) Solve the following system of linear equations using Cramer's Rule and check the correctness of your answer. 4xyz 1 2x + 2y + 3z = 10 5x-2y-2z = -1 b) Find the adjoint of the following matrix A. Then use the adjoint to find the inverse of A if possible, and check the correctness of your answer. A = c) Determine whether the following points are collinear. Why or why not? If not,…

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