Does any one know of a particular textbook or reference that proves existence and uniquence of the ODE $\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x}=f(x,y)$?
Edit: Consider the initial value problem:
$\frac{dy}{dx}=f(x,y)$, $y(x_0)=y_0$ (E)
Assume $f:D\to\mathbb{R}$ is a continuous where $D=\{(x,y):m\leq x\leq n, p\leq y\leq q\}$. Assume that $\phi(x_0)=y_0$, $y_0\in[p,q]$. Then $y=\phi(x)$ is a solution of (E) if and only if
$\phi(x)=y_0+\int_{x_0}^x f(t,\phi(t))dt$.