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When does the IVP

$$\begin{cases}\dot{y}=y^2 \\ y(0)=1,\end{cases}$$

with $(x,y)\in \Bbb R\times\Bbb R$ have a unique solution?

For

$$\begin{cases}\dot{y}=f(x,y) \\ y(0)=y_0 \end{cases}$$

when do we have a unique solution, no solution, or infinitely many solutions? i want to ask the first IVP have a unique solution in which of the following intervals.. a)($-\infty$,$\infty$) b)($-\infty$,1) c)(-2,2) d)(-1,$\infty$).and how the answer is given option b . please tell me how it comes

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The $\belongs$ command does not seem to be rendering the way you want it to. Do you want the symbol, $\in$? Also by $(R,R)$, do you mean $\mathbb{R}^2$? –  Baby Dragon Sep 25 '13 at 4:08
    
yes. i meant this –  abc Sep 25 '13 at 4:09
    
Tri \in ${}{}{}$ –  leo Sep 25 '13 at 4:10
    
What do you mean by solution? Do you mean a function $x \mapsto y(x)$ defined for $x \in (-\epsilon, \epsilon)$ for some small $\epsilon$? This is usually called a 'local solution'. Or do you mean a solution $y : \mathbb{R} \to \mathbb{R}$, that is, defined for all 'time'? You should look up Picard iteration, which is a typical method for establishing local uniqueness. –  A Blumenthal Sep 25 '13 at 4:19

2 Answers 2

up vote 0 down vote accepted

One can solve $\dot y=y^2$ where $y\ne0$ noting that $\dot y/y^2$ is the derivative of $-1/y$. Integrating this yields that the unique local solution on a neighborhood of $0$ is such that $1/y(x)=-x+1/y(0)$. By inspection, if $y(0)=1$, the maximal solution is $y(x)=1/(1-x)$ for $x$ in $(-\infty,1)$.

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what do you mean by maximal solution.and thanks –  abc Sep 26 '13 at 7:14
    
The solution defined on the maximal interval around x=0 on which a solution exists. –  Did Sep 26 '13 at 7:19

The IVP has a uniaue solution by Picard-Lindelof Theorem.

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please someone help –  abc Sep 26 '13 at 6:55
    
@Mhenni how do you apply the theorem here $f(x,y)=y^2$ is not lipschitz! –  Neeraj Bhauryal Sep 8 at 7:17

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