That boundary condition doesn't make sense to me. The term on the right should be a constant, not a function. Is bC supposed to be bC(x,0)?

If so, then the solution for the y problem will be of the form , and . The boundary condition Y'(1) = 0 tells you that . So take . Then .

Now the boundary condition Y'(0) = bY(0) (if that is what it should be), tells you that . So the eigenvalue equation for is . That has infinitely many solutions, but unfortunately you can't solve them explicitly.