Question

?(?) ≔ { 0, 0 ≤ ? < ? 1, ? < ? < 2? 0,...

?(?) ≔ { 0, 0 ≤ ? < ? 1, ? < ? < 2? 0, ? > 2?} (a) Write ?(?) in terms of window function. (b) Write ?(?) in terms of step functions. (c) Determine the Laplace transform of ?(?). (d) Determine ℒ{1 − cos?}. (e) Use the method of partial fractions to determine ℒ−1 { 1 ?(?2+1)}. (f) Evaluate ℒ{(1 − ??? ? )?(? − ?)}. (g) Use the Laplace transform to solve the initial value problem ?"(?) + ?(?) = ?(?); ?(0) = 0, ?′(0) = 1.

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