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I do not have an idea where to start to solve the following differential equation, so every tip is welcome.

$$y' = \frac{x\sqrt{4+y^{2}}}{y(9+x^{2})} $$

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This differential equation is separable, meaning that you can get all the $x$ terms on one side and the $y$ terms on the other. Rewrite $y'$ as $\frac{dy}{dx}$, separate the $x$ terms from the $y$ terms (including $dx$ and $dy$ respectively), then integrate both sides. That will be a good start. –  Théophile Jul 19 '12 at 0:37
    
@Théophile, i understand how to solve differential equations, it's not my first, but i dont have idea how to separate x and y from this expression, because of its complexity –  sun9 Jul 19 '12 at 0:40
    
The right side of the equality can be written as $\frac{\sqrt{4+y^2}}{y} \frac{x}{(9+x^2)}$. Does this help? –  Javier Badia Jul 19 '12 at 0:42
    
@JavierBadia - thanks, it's a little late here in Europe (2:44AM), therefore slower i think :D –  sun9 Jul 19 '12 at 0:45
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1 Answer

We write your equation as $$ \frac{y}{\sqrt{4+y^2}}dy=\frac{x}{9+x^2}dx $$ or equivalenlently, $$ d(\sqrt{4+y^2})=d((1/2)\ln(9+x^2)). $$ Hence, we get $$ \sqrt{4+y^2}=\frac{1}{2}\ln(9+x^2)+C. $$

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