How to find the transfer function of xy, xy' etc?

[goggles31]

For example, how would you find the laplace transform of xy' in
y''-xy+y=1, y(0)=1, y'(0)=2 ?

 

[Mark44]

For example, how would you find the laplace transform of xy' in
y''-xy+y=1, y(0)=1, y'(0)=2 ?

You don't have xy' in your differential equation.

This table, http://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf, #30, gives the Laplace transform of $t^n f(t)$. In your problem x = t, y = f(t), and n = 1.