Question

If u = log(r), where r^2 = (x-a)^2 + (y-b)^2, and (x-1) and (y-b) are not zero simultaneously, show that d^2u/dx^2 + d^2u/dy^2 = 0.

I first used some of the properties of log and made u = 1/2 * log((x-a)^2 + (y-b)^2)

Then made u = 1/2 * ln((x-a)^2 + (y-b)^2)/ln(10)

I'm fairly confident everything up to this point is right but if not please correct me. I then did partial derivatives with respect to x twice and got something like

d^2u/dx^2 = 1/([(x-a)^2 + (y-b)^2]*ln(10)) - (2x(x-a))/([(x-a)^2 + (y-b)^2]^2*ln(10))

At this point i'm hesitant to go any further cause i do not believe that I will get 0 as an end result.

Answer 1

oh never mind. i looked back and saw that i differentiated wrong on a step