How can I solve this ODE ?
$$y'-xy=e^x\tag*{}$$
My approach :
Here, integrating factor,
$$\begin{align}\mu &=e^{\int (-x)\, \mathrm{d}x}\\ &=e^{-x^2/2}\end{align}\tag*{}$$
So, after multiplying by the integrating factor, the ODE becomes
$$\begin{align}ye^{-x^2/2} &= \displaystyle \int e^x e^{-x^2/2}\, \mathrm dx \end{align}\tag*{}$$
The last integral can't be expressed in terms of elementary functions I think. Is there any way to proceed from here ? If there is any mistake in my method, please point it out.
First of all: what do you define as a non usual integral?
Secondly we can write the integral in terms of the error function:
$$\int\exp\left(x-\frac{x^2}{2}\right)\space\text{d}x=\sqrt{\frac{e\pi}{2}}\cdot\text{erf}\left(\frac{x-1}{\sqrt{2}}\right)+\text{C}\tag1$$