Question

7 Question Six: (5 marks) Solve the following integral equation using the Laplace transform. y(t)-/ sin(2t)y(t-r)dr-3. 0

Answer 1

Consider given integral equation. y (t) - integral^t_0 sin (2 tau) y (t - tau) d tau = 3 By applying Laplace transform both sides, we get L {y (t)} + L {integral^_0 sin (2 tau) y (t - tau) d tau} = 3 L (1) L {y (t)} + L {sin (2 t)} L {y (t)} = 3/s Now use convolution theorem L {y (t)} (1 + 2/s^2 + 4) = 3/s (s^2 + 6/s^2 + 4) L {y (t)} = 3/s L {y (t)} = 3 (s^2 + 4)/s (s^2 + 6) f (t) = L^-1 {3 (s^2 + 4)/s (s^2 + 6)} Consider 3 (s^2 + 4)/s (s^2 + 6) = A/s + B s + C/s^2 + 6 \r\n = A (s^2 + 6) + (B s + C) s/s (s^2 + 6) 3 (s^2 + 4) = A (s^2 + 6) + (B s + C) s Now put s = 0,