1. (a) The signal x(t) from a control system obeys the equation

d²x dx

+8 +7x = 42

dt

dt²

with initial conditions x(0) = 1 and (0) = -1. Use Laplace Transforms to

determine the function x(t).

[12 marks]

(b) The ordinary differential equation

d²h dh

3 +5 + 2h = cos x

dx² dx

is to be solved numerically over the domain 0 ≤ x ≤ 1, which is represented by

N equally spaced nodes, with node 1 located at position x = 0.

(i) Using second order accurate finite difference approximations, derive a

general expression for the value of h at node i + 1 in terms of its values at

[6 marks]

nodes i and i - 1.

dh

(ii) Show how the boundary conditions h = 2 and d = 3 at x = 0 could be

incorporated into the numerical solution process and write down the finite

difference equations that apply at nodes 1, 2 and 3.

[5 marks]

(iii) Explain briefly how the numerical solution process would need to change if

boundary conditions were instead applied at both ends of the domain.

[2 marks]