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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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    @Willie: Yes, I know. I was just wondering what exactly Vafa's question is.2011-03-07

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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.2011-03-07
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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:

http://www.amazon.com/Partial-Differential-Equations-Mathematical-Sciences/dp/0387906096/ref=sr_1_1?s=books&ie=UTF8&qid=1299461183&sr=1-1

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.