Question 1
(a) Using Λf(t)'] = sF(s) - f(0) , show that Λ [sin2(3/2t)] = 9/2s(s2 + 9)
(b) (i) If f(t) has the Laplace transform F(s), then show that
where u(t) is a step function.
(ii) Find the inverse transform of se-3s/(s2 +4)
(c) Find the Laplace transform of f(T) = e-3t (cost - 3sint). Express it in a simple compact form.
Question 2
(a) Using the Laplace transform method solve the following differential equation:
(b) Express the following function in terms of unit step functions and then find its Laplace transform:
(c) Applying the convolution theorem, find h(t) = L-1{H(s)} for
H(s) = s2 /(s2 + 4) (s2 + 9)
Question 3
(a) Using the technique of Laplace transformation find the solution of the following set of differential equations:
(b) Verify if the function v(x,y) = xy is a harmonic one. If yes then first find its harmonic conjugate u(x,y) and then find the analytic function f(z) = u(x,y) + iv(x+y) as a function of the complex variable z.
(c) Using Cauchy-Riemann equations verify if the following function is analytic:
