Find the laplace transform of the function obtained and

Question

Question 1:

(a) Find the Laplace transform of the following function:

f(t) = e^-tsin2t

(b) Given the following function as:
f(t) = 2t if 0 < t < 3
      = 0  if t > 3

(i) Express this function in terms of unit step functions.

(ii) Find the Laplace transform of the function obtained in (i).

(c) Find the inverse Laplace transform of the following function:

F(s) = s-3/4s² -24s +39

Question 2:

(a) Find the Laplace transform of f(t) = cosh 2t sin 4t.

(b) Using Laplace transforms, solve the following differential equation:

y'' + 9y = 1, y(0) = 0, y'(0) = 4.

(c) Using the concept of determining the inverse Laplace transfonn from the derivative of a function, i.e., (L^-1(-F'(s))=t f(t)), find the Laplace transform of the following function:

1/2te^-2t sint

Question 3:

(a) Verify if the function u(x,y) = x² - y² is a harmonic one. If yes then first find its harmonic conjugate v(x, y) and then find the analytic function f(z) = it(x, y) + iv(x,y)as a function of the complex variable z.

(b) Using the Laplace transformation, solve the following integral equation:

y(t) + 2e^t ₀∫¹ y(τ)e-^τdτ = te^-t.

(c) Find the value of the following integral for C:|z| = 1 anticlockwise

∫_c e^3z/(4z-Πt)^{3 dz}