Linear First Order ODE/y' + 2 x y = exp -x^2

Theorem
The linear first order ODE:
 * $\dfrac {\d y} {\d x} + 2 x y = \map \exp {-x^2}$

has the general solution:
 * $y = \paren {x + C} \, \map \exp {-x^2}$

Proof
$(1)$ is in the form:
 * $\dfrac {\d y} {\d x} + \map P x y = \map Q x$

where:
 * $\map P x = 2 x$
 * $\map Q x = \map \exp {-x^2}$

Thus:

Thus from Solution by Integrating Factor, $(1)$ can be rewritten as:
 * $\map {\dfrac {\d} {\d x} } {\map \exp {x^2} y} = \map \exp {-x^2} \, \map \exp {x^2} = 1$

Hence: