Implicit Integration Help

Question

$\frac{dy}{dx}+2ty=e^{-t^2}$, solve for $y$.

attempt

$e^{\int2t}=e^{t^2}$

$e^{t^2}\frac{dy}{dx}+2te^{t^2}y=e^{t^2}e^{-t^2}$

$\int \frac{d}{dx}(e^{t^2}y)=\int 1$

$e^{t^2}y=x+c \implies y=\frac{x}{e^{t^2}}+\frac{c}{e^{t^2}}$