For a time-dependent function f(t) and the corresponding Laplace transform F(s), show that the following identities are correct: (i) L(t f(t)) = _ dF(s) ds (ii) c(f()) = a F (as) [6 marks] (c) Consider the following differential equation: x(t) + x(t) = te-³t with x(0) = 1 (1) Using the result of part (a) and the identities derived in part (b), calculate the Laplace transform of t e-³t [2 marks] (ii) Laplace-transform the differential equation and calculate the solution to the differential equation in the s-domain, X(s). [4 marks] Calculate the solution to the differential equation in the t-domain, x(t), via inverse Laplace transform. [5 marks]