How do I go about applying the operator C to the function? Do i take the derivative of 1/r^2 wrt r, then multiply by the derivative of r^2 wrt r + 2/r. Then all of this multiplied by the function Y?

Yea, this notation is confusing. First, about the +2/r part. The result of C*Y (C operating on Y) is (2/r)*Y, and is added to the result of the other part.

Now for the (1/r

^{2}) (d/dr) (r

^{2} * (d/dr)) part... You put Y in at the end, so this means (1/r

^{2}) (d/dr) (r

^{2} * (d/dr)Y), in other words, take (d/dr) of Y, then multiply by r

^{2}, then take (d/dr) of this result, then multiply by 1/r

^{2}.

Or, you can expand out the whole operator. You get:

(1/r

^{2}) * (2r * (d/dr) + r

^{2} * (d

^{2}/dr

^{2})) + (2/r) =

d

^{2}/dr

^{2} + (2/r) * (d/dr) + (2/r)

To make this halfway readable, write Y' for dY/dr, etc.

Then, C*Y = Y'' + (2/r) Y' + (2/r) Y

which is now halfway sane to read.