I want to solve $$\frac{dy}{dx}=e^{x^{2}}.$$ i using variable separable method to solve this but after some stage i stuck with the integration of $\int e^{x^{2}}dx$. i dont know what is the integration of $\int e^{x^{2}} dx$. Please help me out!
Solve $dy/dx = e^{x^2}$
5
$\begingroup$
ordinary-differential-equations
-
0There is no closed form integral of $e^{x^2}$. Are you sure you don't have a typo? – 2012-07-20
-
3@Ed: it has a closed form. It's not *elementary*, however. – 2012-07-20
-
0Er, yes. Thanks for that correction. I should have been more precise. – 2012-07-20
-
0See, for instance, [this](http://math.stackexchange.com/questions/138664) or [this](http://math.stackexchange.com/questions/155). – 2012-07-20
-
1Why in this green earth was this question downvoted? – 2012-07-20
-
1@J.M. Maybe someone was on the blue part of Earth. – 2012-07-20
-
0@J.M. Honest question: what does *closed form* mean if not *elementary*? – 2012-07-20
-
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
3
$$ \frac{dy}{dx}=e^{x^{2}} $$ 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. ...?
-
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