First Order ODE/y' + 2 x y = 1

Theorem
The first order ODE:
 * $y' + 2 x y = 1$

has the general solution:
 * $y = e^{-{x^2} } \ds \int_a^x e^{t^2} \rd t$

where $a$ is an arbitrary constant.

Proof
This is a linear first order ODE with constant coefficents in the form:
 * $\dfrac {\d y} {\d x} + a y = \map Q x$

where:
 * $a = 2 x$
 * $\map Q x = 1$

From Solution to Linear First Order Ordinary Differential Equation:
 * $\ds y = e^{-\int P \rd x} \paren {\int Q e^{\int P \rd x} \rd x + C}$

Thus

Further work on this is not trivial, as $\ds \int e^{x^2} \rd x$ has no solution in elementary functions.