System of non-linear differential equations $\alpha_1 g(f-g)+\beta_1 f''=0$ and $\alpha_2 f(f-g)+\beta_2 ​g''=0$.

Question

Let $\alpha_1,\alpha_2,\beta_1,\beta_2\in\mathbb{R}-\{0\}$. Consider the system of Differential equations given by $$\alpha_1 g(f-g)+\beta_1 f''=0$$ $$\alpha_2 f(f-g)+\beta_2 g''=0$$ I want to find all the twice differentiable functions $f,g:\mathbb{R}\to\mathbb{C}$ that satisfies above equation. I got this when I was studying about electron dynamics in theoretical physics.