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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 $f$? – Willie Wong Mar 7 '11 at 1:20
@Willie Wong: I have edited the questiion will all the details. – Vafa Khalighi Mar 7 '11 at 2:16
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. – Hans Lundmark Mar 7 '11 at 7:47
@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. – Willie Wong Mar 7 '11 at 13:03
@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. – Willie Wong Mar 7 '11 at 13:07

The intro ODE text by Boyce and DiPrima gives a fairly complete outline of this proof. Some of the details are left as exercises for the reader, but since the text is aimed at an introductory audience, anyone who knows enough to ask this question would presumably have little trouble filling in the gaps.

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I presume you mean existence and uniqueness of the solution of this ODE? It seems you're referring to the Picard–Lindelöf theorem? That's proved in the corresponding Wikipedia article.

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Thanks but I need a complete proof that is easy to follow. – Vafa Khalighi Mar 7 '11 at 2:18

The usually way they do this is via some kind of contract mapping theorem. This theorem is called the fundamental theorem of ODE, it can be extended to $n$ variable situation via similar arguments.

The book I learned this in high school is probably this one:

I think you can find similar ones in any standard ODE books. In particular Arnold's book has an incomplete proof based on some assumption in the very start.

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