I started my first pde course. I don't have much experience with this yet.
I found an interesting question: Solution to one-dimensional Wave Equation with Method of Characteristics, but I didn't understand four points. I wish someone could help me.
First: In the Sy 1, he fixed the vectors tangents to the curve $\Gamma$?
Second: Can $\Gamma$ be any smooth curve?
Third: why $F(c_1,c_2) = 0$? who's $F$? I have no idea
Fourth: why $u = g(y-c_{0}x)$? Here, I tried to use the implicit function theorem, but I don't know what it would be like to calculate $F_u$.
The points first, three and four are the main problems.
For second, I think it can really be any smooth curve. I didn't seem to use properties of any particular curve.
Now, my question:
After, using the same ideia I will to got $u=f(y + c_{0}x)$. We know that the solution is $u(x,y) = f(y + c_{0}x) + g(y - c_{0}x)$... why the solution is the sum of the solutions to two one-dimensional equations?