Can we deduce anything about existence of a solution of a differential equation using Implicit function theorem. I feel we can but I am unable to setup things to apply implicit function theorem.
Let's consider the simplest case, we want to solve $y'=f(x,y)$ on $\mathbb R.$ For this I was trying to define a map $f:\mathbb R\times\mathbb R^2\to\mathbb R^2$ by $$f(x,y,\dot y)=(\dot y-f(x,y),g(y,\dot y))$$where $g$ is such a function which basically implies $\frac{dy}{dx}=\dot y.$ And using Implicit function theorem, we will $y=h(x)$ which solves the differential equation. I know my writing is ambiguous, I don't know what to write. Please help to correct things and give some idea how should I define $g$. Thank you.