Yes, running 10.02 on Windows 8.1

I usually break it up to make it more readable:

ClearAll["Global'*"]

eqns = {D[u[x, y], x, x] + D[u[x, y], y, y] - 3 u[x, y] == x y, 
  u[0, y] == 0, (D[u[x, y], x] /. x -> 2) == 0, u[x, 0] == 0, u[x, 1] == 0}

Traditional form lets me check if my equations make sense!

TraditionalForm[eqns]

s = NDSolve[eqns, u, {x, 0, 2}, {y, 0, 1}]

Plot3D[u[x, y] /. s[[1]], {x, 0, 2}, {y, 0, 1}]

Thank you for your help but you misunderstood me. My teacher said that I need to figure it out by myself not because I need to learn how to solve such problems but because he just doesn't know what is wrong and can't help me as he is a mechanical engineering (not math) expert. And if he - Doctor of Engineering can't solve it then how I - second year student can do it ?

What's more he suggested that I should either ask on some forum or leave this problem and continue with the rest of my project, becuase it's very small part of my finite element analysis project and it's only needed to check if whole thing is correct. I don't have time to sweat blood to find solution.

Also complex differential equations are not important part of my studies as I study biomedical engineering, not mathematics.

So could you please tell me where is the mistake in my formula ? I would be very grateful.