Question #ee7ad

1 Answer
Jim H
Apr 7, 2017

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#.

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