Question

Previous Problem Problem List Next Problem (1 point) In this exercise we will use the Laplace transform to solve the following initial value problem: 0s(O)1 (1) First, using Y for the Laplace transform of y(t). i.e., Y-L(y(t), find the equation obtained by taking the Laplace transform of the initial value problenm 2) Next solve for Y - (3) Finally apply the inverse Laplace transform to find y(t) y(t) =

Answer 1

Given y' + y = { -1 0 0 lessthanorqualto t lessthanorqualto1 1 lessthanorqualto t, y(0) = 1 (1) consider g (t) = { -1 0 0 lessthanorqualto t lessthanorqualto 1 1 lessthanorqualto t now to find L { g(t) } we do this by definition: Y (S) = Integral^infinity_0 e^-st (g (t)) = integral^1_0 (-1) e^-st dt + integral^infinity_1 (0) e^-st dt = integral^1_0 (-1) e^-st dt = (e^-st/s)^1_t = 0 = -1 + e^-s/ s (2) now consider y' + y = { -1 0 0 lessthanorqualto t lessthanorqualto 1 1 lessthanorqualto t, y(0) = 1 L { y' + y } = L { g(t) } Rightarrow sY (s) - y (0) + Y (s) = -1 + e^-s/s