Let $f_1, f_2: \mathbb R \to \mathbb R$ be two smooth functions. I now want to show that the vector fields $X_1, X_2$ on the $\mathbb R^2$, given by
$$X_1(x, y) := f_1(y) \partial_x \qquad \text{ and } \quad X_2(x, y) := f_2(x) \partial_y$$
are complete.
Now I think to show completeness, I need to calculate the flow for the integral curve and show that it's maximal? So I tried to write down the differential equation
$$\dot \gamma = X_1 \circ \gamma \tag{1}$$
for the first vector field; I think I'd need to solve this for the integral curve $\gamma$ and show that $\gamma$ is maximal, i.e. it can be defined on all of $I = \mathbb R$?
But I'm not really sure hwo to continue here. By writing $\gamma(t) = \begin{pmatrix} \gamma_1(t) \\ \gamma_2(t) \end{pmatrix}$, I tried to write out (1) as
$$\dot \gamma = f_1(\gamma_2) \partial_{\gamma_1(t)}$$
But I'm a bit lost on how to solve this and if I'm even on the right track here.