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I would like to solve the following differential equation

$$ y' = \alpha e^{-y^2} $$

How should I proceed ?

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It's fairly well known that $\int e^{-x^2}$ can't be expressed in terms of elementary functions. – Robert Mastragostino Jun 7 '12 at 17:33
$y$ is the unknown function. I did not write $e^{-x^2}$. Edit : My bad I wrote ODE, which was confusing. I edited the post. – vanna Jun 7 '12 at 17:37
Just note that $dx/dy = \alpha^{-1}e^{y^2}$. – Sangchul Lee Jun 7 '12 at 17:44
You mean that I should find $y^{-1}$ and invert the result to get $y$ ? So as expected a non-closed formula... $y(x) = (\int_{y_0}^. \exp(u^2)du)^{-1}(x)$. Is that it ? – vanna Jun 7 '12 at 18:05
up vote 1 down vote accepted

Solve the differential equation as x is a function in y. This gives the answer:

$$ x={\frac {1/2\,i\sqrt {\pi }\,{{\rm erf}\left(iy\right)}}{\alpha}} + C $$.

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