I am working through Forster's Analysis II and am finding it difficult to understand the formulation of ordinary differential equations :
Let $G \subset \mathbb{R} \times \mathbb{R}^{n}$ be a subset, and $f:G \rightarrow\mathbb{R}^{n} \quad (x, y) \rightarrow f(x, y)$ be some continuous function. Then $y' = f(x, y)$ is a system of n differential equations of order 1, and the solution on some interval $I$ takes the form $$ϕ'(x) = f(x,ϕ(x)) \quad \forall x ∈ I.$$ I am confused about f being a function of $x$ and of $\phi(x)$. Is it not then, just a function of $x$ ?
I am in particular confused about the Lipschitz condition : for some $L \geq 0$ $$||f(x, y) - f(x, \tilde{y})|| \leq L ||y - \tilde{y}|| \quad \forall (x, y), (x, \tilde{y}) \in G$$ Since I expect $y$ and $\tilde{y}$ are still functions of $x$, I am confused as to how $y$ and $\tilde{y}$ differ without a change in $x$. Should I understand $y$ and $\tilde{y}$ to be members of the family of solutions to the differential equations which differ by some constant ? Is the Lipschitz condition then a statement on how drastically changing this constant affects f ?
In both definitions (of the system of differential equations and Lipschitz condition), $x, y$ should be considered as independent variables (more precisely, $y$ is an $n$-vector, so we actually have n+1 independent variables). But if there is a function $\phi = \phi(x)$, we can substitute $y$ with $\phi(x)$. Of course, after such a substitution, $f$ becomes a composite function of one variable $x$.