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Use the existence and uniqueness theorem to prove that $y(x)=3$ is the only solution to the IVP $$y^{\prime}=\frac{x}{x^2+1}(y^2-9), \,y(0)=3$$ I have no idea on how to start. How to show the existence of a solution ?

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You're told to use the existence and uniqueness theorem, so a good place to start would be to recall the statement of the existence and uniqueness theorem. Then prove that $$f(x,y) = \frac{x}{x^2 + 1}(y^2 - 9)$$ satisfies the hypotheses of the existence and uniqueness theorem. – Henry T. Horton Jan 28 '13 at 17:03
You are told what the solution is. You must prove that it is the only solution. – Julián Aguirre Jan 28 '13 at 17:15
Let $f(x,y)=\frac{x}{x^2+1}(y^2-9)$. Then both $f$ and $f_y=\frac{2x}{x^2+1}(y-9)$ are continuous on $\mathbb{R}^2$, there exists a unique solution. Then we just have to show that $y(x)=3$ is indeed a solution to the DE above . Is it okay ? – Idonknow Jan 29 '13 at 10:01

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