1. The problem statement, all variables and given/known data Please take a look at the example at the bottom (at eq. 18:18, at page 619): http://books.google.dk/books?id=9p6...g=iAUGf-f9d5uHRwwYZnpFpA8HJDA&hl=da#PPA619,M1 Q1: In case (ii), why do they add g(x^2+y^2)? Q2: Why do they not add it in case (i)? Q3: In case (ii), if I chose f(p) = p + 1 instead of f(p) = 2p, then the solution would be u(x,y) = 1+ x^2 +y^2 + g(x^2 + y^2), right? I hope you can help me; I find this really hard to understand, and I've spent hours trying to find out, but I find that the book is poorly written. They don't emphasize the important things at all, and the reader is left behind with so many questions. Thanks in advance.
The difference between (ii) and (i) is that in (i) it is a function, but in (ii) it is a single point. Although I am not sure about this?
The difference between i) and ii) is that in i) the boundary conditions are given along a line y=0 and in ii) they are specified only a a single point. In the first case that's enough information to specify a unique solution. In the second case you can add a general function that solves the homogeneous equation (g(x^2+y^2)) subject to only one condition. I don't see why they are messing around with splitting g into two parts. That is confusing. I would just write the solution as -3y+g(x^2+y^2) subject to the condition g(1)=2. Your other form for Q3 is fine, but you forgot to put the -3y in it.