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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

$\phi(x)=y_0+\int_{x_0}^x f(t,\phi(t))dt$.

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    What differentiability/continuity conditions do you assume on $f$?2011-03-07
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    @Willie Wong: I have edited the questiion will all the details.2011-03-07
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    The edited question doesn't seem to ask the same thing as the original one. What you're asking now is simply how to reformulate the *differential* equation as an *integral* equation, right? This is basically just the fundamental theorem of calculus – a much simpler step than actually proving existence and uniqueness of the solution.2011-03-07
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    @Hans, the re-writing of the differential equation as an integral equation is usually the first step to proving existence theorems using an iterative scheme. But you are right, it is not the whole picture.2011-03-07
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    @Vafa: the point I asked is that, if $f$ were merely continuous (which you postulates above) and not Lipschitz, uniqueness to the solution of the ODE can fail. There is an example near the bottom of the Wikipedia page that Joriki linked to below. Also, generally one prefers that the domain $D$ is open, or that $(x_0,y_0)$ is an interior point, else you can't necessarily get started with the iteration argument.2011-03-07
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    @Willie: Yes, I know. I was just wondering what exactly Vafa's question is.2011-03-07

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