Definition of the laplace transform

Question

Ordinary Differential Equations - The Laplace Transform

Explain the steps on how to solve the problem below:

a) Use the definition of the Laplace transform :
to find F(s) if          

    F(S)=∫^∞₀ ƒ(t)e^-tsdt
    ƒ(t)=3e^-3t-5sin2t

b) Use the properties of the Gamma function to calculate Γ (7/2).

c)Use the Laplace properties and the s-shifting theorem to solve the differential equation

        yⁿ-4y′+8y=2e^-2t

subject to inintial condition y(0)=1/10,y′(0)=1/5