I want to solve $$\frac{\, \mathrm dy}{\, \mathrm dx}=e^{x^{2}}.$$ i using variable separable method to solve this but after some stage i stuck with the integration of $\int e^{x^{2}}\, \mathrm dx$. i dont know what is the integration of $\int e^{x^{2}}\, \mathrm dx$. Please help me out!
Solve $\, \mathrm dy/\, \mathrm dx = e^{x^2}$
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ordinary-differential-equations
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1@Rahul: I wrote [a long-ish answer to that question](http://math.stackexchange.com/a/9203) many moons ago, but the gist is, I consider the error function as a "known quantity", and I thus treat it as a closed form. – 2012-07-20
1 Answers
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$ \frac{dy}{dx}=e^{x^{2}} $ has no elementary solution. The error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined as: $ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt. $ See the link for reference and more information and thus, J.M. ...?
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3...and thus, the solution of the DE is $y=-\frac{i\sqrt\pi}{2}\mathrm{erf}(ix)+C$. (the *imaginary error function*) – 2012-07-20