Question

Use the properties of Laplace transforms that we review in class, along with the table of Laplace transforms that is posted on Canvas under the module Supplementary Material,to answer the following questions. (a) Let f(t) = cos(wt), g(t) = 20 sin(at) (t ≥ 0), where @ and a are real numbers.Compute the Laplace transform of the convolution f(t) * g(t) (t ≥ 0). (b) Find the Laplace transform F(s) of the following function, where 8(t) is the Dirac delta function. You may leave your answer as a sum and do not need to manipulate the expression to get a common denominator. f(t) = 38(t) + (t + 1)² + e−²t sin(3t) (t > 0) (c) Apply the Laplace transform to the following ODE and find an expression for Y(s),the Laplace transform of y(t), as a quotient of two polynomials in s. The function us(t)is the unit step function. \frac{d^{2} y(t)}{d t^{2}}+3 \frac{d y(t)}{d t}+2 y(t)=u(t) \quad y\left(0^{-}\right)=-1, \quad \dot{y}\left(0^{-}\right)=2, \quad u(t)=10 u_{s}(t)

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