Derive the laplace transform and the inverse laplace

Question

Q 1. Derive the Laplace transform of the sketched f(t) below:

Q 2. Find the inverse Laplace transform of the following functions:

(i) s/L²s² + n²Π²

(ii) 1/(s + √2)(s - √3)

Q 3. Using Λ[f(t)'] = sF(s) - f(0) , show that Λ[sin²t] = 2/ s(s² + 4)

Q 4. Solve the following initial value problem by the Laplace transform:

y'' + 2 y' - 3y = 6e^-2t, y(0) = 2,

y''(0) = -14.

Q 5. (a) If f(t) has the Laplace transform F(s), then show that

Λ{ f (t - a)u(t - a)} = e^-as F(s),

where u(t) is a step function.

(b) Find the inverse transform of e^-3s /s³.

Q 6. Using the Laplace transform, solve the following problem.

y'' + 2 y'' - 3y = 8e^-t + δ (t - 1/2), y(0) = 3, y'(0) = -5.

Q 7. By differentiation, find the inverse Laplace transform of:

s² - Π²/(s² + Π²)²

Q 8. By differentiation, find the Laplace transform of:

t² coshΠt

Q.9 Using Laplace transforms solve the following integral equation:

y(t) = sin 2t + ₀∫^t y(t') sin 2(t - t')dt'

Q. 10: Evaluate the following integral:

_C∫2z³ + z² + 4/ z⁴ + 4z² dz,

where C is the circle |z - 2| = 4 clockwise.