TITLE: Solve $\, \mathrm dy/\, \mathrm dx = e^{x^2}$
QUESTION [7 upvotes]: 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!

REPLY [4 votes]: $$
\frac{dy}{dx}=e^{x^{2}}
$$
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. ...?