Mathematical Methods for Physics and Engineering - Matematica.NET

13.4 EXERCISES(c) L [sinh at cos bt] = a(s 2 − a 2 + b 2 )[(s − a) 2 + b 2 ] −1 [(s + a) 2 + b 2 ] −1 .13.24 Find the solution (the so-called impulse response or Green’s function) of theequationT dxdt + x = δ(t)by proceeding as follows.(a) Show by substitution thatx(t) =A(1 − e −t/T )H(t)is a solution, for which x(0) = 0, ofT dxdt + x = AH(t), (∗)where H(t) is the Heaviside step function.(b) Construct the solution when the RHS of (∗) is replaced by AH(t − τ), withdx/dt = x =0fort0, the function y(t) obeys the differential equationd 2 ydt + a dy2 dt + by = c cos2 ωt,where a, b and c are positive constants. Find ȳ(s) andshowthatsȳ(s) → c/2bas s → 0. Interpret the result in the t-domain.13.26 By writing f(x) as an integral involving the δ-function δ(ξ − x) and taking theLaplace transforms of both sides, show that the transform of the solution of theequationd 4 ydx − y = f(x)4for which y and its first three derivatives vanish at x = 0 can be written as∫ ∞ȳ(s) = f(ξ) e−sξ0 s 4 − 1 dξ.Use the properties of Laplace transforms and the entries in table 13.1 to showthaty(x) = 1 ∫ xf(ξ) [sinh(x − ξ) − sin(x − ξ)] dξ.20465