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Solving a problem which relates to the movement of a charged particle in an electric field I had to solve the following diff-equation:

$$y\frac{dy}{dx}=-\frac{a}{x}+by$$ where $(a,b>0)\,\text{and}\,y(x_0)=0;x_0>0$

Wolfram Alpha is not able to solve it.

Any hint?

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    $yy'=\frac12 (y^2)'$, not sure if it helps...2012-08-31
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    Why are you expecting it to be solvable?2012-08-31
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    @Sasha Is there any reason to believe otherwise?2012-08-31
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    at least approximately.2012-08-31
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    @Sasha: Doesn't the Picard–Lindelöf theorem tell us that there will always be a unique local (possibly global) solution to any first order ODE with initial conditions?2012-08-31
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    @FlybyNight Sorry, I misspoke. I meant to ask "why are you expecting it to be solvable in closed-form".2012-08-31
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    A solution in terms of special functions is also a closed form for me2012-08-31
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    I think I've found a solution. See below.2012-08-31
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    Similar to http://math.stackexchange.com/questions/14638012016-05-22

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