Question

Question #ee7ad

Answer 1

I assume that you want the harmonic conjugate. I believe that `v(x,y) = 2xy-2y+C` (But it's not my field. I had to review how to find a harmonic conjugate.)

Explanation:

First note that

`(del^2u)/(delx)^2 + (del^2u)/(dely)^2 = 2+(-2) = 0` so `u(x,y)` is harmonic.

We need `u_x = v_y`

`u_x=2x-2 = v_y`.

Now integrate (partially) with respect to `y` to get

`v = 2xy-2y+h(x)`

To find `h(x)` use `-u_y = v_x`, so

`-u_y = -(2y)` and `v_x = 2y+h'(x)` .

Therefore `h'(x) = 0` so `h(x) = C` for some constant `C`

Finally then

`v(x,y) = 2xy-2y+C`.