I've been trying to solve this one problem for days. Literally. Days. This is my method of last resort, so I'm praying someone can explain this to me. I understand the method of characteristics, separation of variables, etc. What is SERIOUSLY do not understand is how to apply these techniques when the PDE is equal to a variable. The problem I am working on is as follows: $$u_x + (\sin x)u_y = y, \; u(0,y) = 0$$
What I have done is $\frac{dy}{dx} = \sin x \rightarrow dy = \sin x dx \rightarrow y = -\cos x + C \rightarrow C = y + \cos x$. Then, we have $u(x,y) = f(y + \cos x)$. Following from this, we should have $0 = f(y + 1)$ which makes no sense to me. I might be missing something extremely obvious because at this point my brain is absolutely fried from looking at this problem for so long.
This is my first post here so I apologize if I have made any rules/procedures faux pas.
ETA: I should also note I know the original equation can be manipulated to become $u_x + (\sin x)u_y - y = 0$ but I have even less of a clue what to do with that straggling $y$ now.
