Suppose that we have a function $f(x)$ of a real variable $x$ in some domain of $U\subseteq\mathbb R$. Also, suppose that this function satisfies the ODE $$(f'(x))^2=4f^3(x)-g_2f(x)-g_3\,,$$ which is the ODE that defines the Weierstrass elliptic function. This function is generally defined over the complex numbers and is also complex.
My question is the following: Since we assume that the variable $x$ is real, is the solution to the above ODE still the Weierstrass elliptic function? If yes, is this solution real itself?