Please see pic 1 for the question and pic 2 for Proposition 2.1. Thank you. 44. Suppose that f is a continuous function for t > 0 and isof exponential order.(a) If f(t) → F (s) is a transform pair, prove thatF (s){/ f(t)dt}(s) =tSHint: Let g(t) = S f(t)dt. Then note that g'(t) =f (t) and use Proposition 2.1 to compute L{g'(t)}(s).(b) Use the technique suggested in part (a) to find1L-1s (s² + 1) ] 5.2 Basic Properties This section will discuss the most important properties of the Laplace transform.of the Laplace We will state each property in a proposition.TransformThe Laplace transform of derivativesThe most important property for our purposes is the relationship between the Laplacetransform and the derivative. As we will see, the next proposition is the key toolwhen using the Laplace transform to solve differential equations.PROPOSITION 2.1Suppose y is a piecewise differentiable function of exponential order. Suppose alsothat y' is of exponential order. Then for large values of s,L(y')(s) = s L(y)(s) – y(0) = sY (s) – y(0),(2.2)where Y (s) is the Laplace transform of y.

Name: Please see pic 1 for the question and pic 2 for Proposition 2.1. Than
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Description: Please see pic 1 for the question and pic 2 for Proposition 2.1. Thank you. 44. Suppose that f is a continuous function for t > 0 and isof exponential order.(a) If f(t) → F (s) is a transform pair, prove thatF (s){/ f(t)dt}(s) =tSHint: Let g(t) = S