solution of $f'(x)+f(-x)=e^{-x^2}$

Question

let $f$ be a function such that $f:\mathbb{R}\to \mathbb{R}$, I want to determine all functions of class $C^1$ such that $f'(x)+f(-x)=e^{-x^2}$ for all $x\in \mathbb{R}$, Now we have $f'(x)+f(-x)=e^{-x^2}$ this implies that $f'(x)=e^{-x^2}-f(-x)$, since $f'$ is of class $C^1$ this means that $f$ is of class $c^2$, That equation equivalent to : $f''(x)=-2x e^{-x^2}-f'(-x)$ implies to $f''(x)-f(x)=-2x e^{-x^2}-e^{x^2}$ , if I take now $x\mapsto \lambda\cos(x)+\mu\sin(x),\lambda,\mu\in\mathbb R$ as a solution without second term in order to get particular solution it would be complicated and given by error function which forbide me to get a general solution, I ask now if there is any simple way to solve that functional ?