solve please complete with all parts please quickly i need it

(a) (b) (C) (32 +4)2' (ii) Use Laplace transforms to solve the differential equation (i) Find the function f (15) whose Laplace transform is y" + 43; = sin(2t) with initial conditions y(0) = a and y'(0) = b, where a and b are real numbers. Sketch the region of integration in the integral 1 1 3 f / 2:56\" dy dx. 33:0 y=m3 Evaluate this integral. Let 2 K($,t) = 65\" t_$(:r:3t + $2 + 1). Show that K(:1:, t) is dominatedly convergent for a: 2 1 and t E [0, 3]. m w