You can use these two beauties:
Changing the cartesian coordinates system to hyperbolic system leads to a very simple PDE. So, the general solution is obtained (in attachment).
Then the condition u(1,y)=1+y will be easy to apply in order to express the particular solution.
I quite not understand your question
x = rho*cosh(theta) and y = rho*sinh(theta)
The inverse is : rho = sqrt(x²-y²) and theta = argtanh(y/x)
This is the usual relationship between cartesian coordinates and hyperbolic coordinates.
- shooting fish in a barrel
I case study new method, which seems easier. I've done this example, but I got the another answer.
After that I talled t and L
If you understand this method, plese give me the answer it is correct?
Sorry, in fact, I didn't read your method.
I just bing back your result u(x,y) into the PDE. It doesn't aggree.
Clue : The final solution of your problem is very simple : u(x,y) = x+y