Task:
Find the function $f(x,y)$ that solves the differential equation $$2\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}=0$$ that also satisfies the condition $f(x,0) = sin (x)$.
My attempt at a solution:
So I have with variabel substitution $u=x-2y, v=x+2y$ solved the equation for the entire plane. I get $$f(x,y)=g(x-2y).$$
I am having some trouble finding the function that also satisfies the condition.
Since $f(x,y)=g(x-2y)$ then $f(x,0) = g(x)$. Now I somehow want to put $g(x) = sin (x)$ and call it done. But the answer in my textbook is $f(x,y) = sin(x-2y)$.
How should I be reasoning when solving this problem?
$g (x)=\sin \, x$ is correct and $f(x,y)=g(x-2y)=\sin (x-2y)$. Your job is not done when you say $g (x)=\sin \, x$ because you are asked to find $f$.