I'm unsure if I've managed to get this right here. Taking the Fourier transform of the entire thing, I get $$2Y(k)+Y(k)G(k) = G(k)$$ Since $\int_{-\infty}^{\infty}g(x-t)y(t)dt$ is the convolution of $y(x)$ and $g(x)$. Solving for $Y(k)$ in terms of $G(k)$ I get $$Y(k) = \frac{G(k)}{2+G(k)}$$
Is the answer really this simple? This just seems too easy, is there something I'm missing?