Laplace transform of squared Function and a multiple of several functions?

I stumbled upon the following differential equation:

$y'=k_1(k_2-y)y$

I've never encountered a squared function before so I'm kind of stumped on how to proceed with this. I tried putting the following into Wolfram Alpha:

$\mathscr{L} \{(f(t))^2 \}$

Which gave me the following error:

(no result found in terms of standard mathematical functions)

So I could use some help with that as Googling didn't give me what I was looking for. On the topic I was also curious about rules for laplace transform of two functions multiplied, I know that the laplace transform is a linear operator but maybe if there's some sort of general rule that applies here:

$\mathscr{L} \{f(t) \cdot g(t) \}$

I'm not expecting someone to spoonfeed me too much here but a pointer to some site or pdf/book would be nice. I'm not the most informed on Laplace Transforms though, so try to keep it somewhat simple :) Thanks