Find Laplace transform for each of the following functions, f(x) = 3e^(2x)+x^(2), f(x) = sin(3x+2)

solve (x^2y"-xy'+y=(x^2+1) using laplace transforms
Please show all your work, and do not use eulers technique once the first Laplace transform has been applied, as that is not what I am curious about. However if it is not possible to solve this particular ODE using only Laplace transforms in combination with other ODE solving techniques (once the Laplace transform has been applied). Then please let me know

Is there any physical significance of laplace transform? 

let f(x)=5x-2 and g(x)=3x+1. write a formula for each of the following functions and then simplify.
a. f+g=
b. f-g=
c. g-f=
d. g+f=
e. fg=
d. ff=