We know that the standard metric used in the regular 2D Euclidean plane is:
$$ds^2=dx^2+dy^2=dr^2+r^2d\phi^2$$
and the shortest path curves (i.e. Geodesics) are simply straight lines spanning all the 2D plane. I tried to change the metric a bit to obtain
$$ds^2=g(r)[dr^2+r^2d\phi^2]$$
with $g(r)\ge 0$, which preserves the rotational symmetry of the 2D plane, but warps it. By calculations, the new differential equation of the shortest path curves becomes
$${C_1dr\over \sqrt{{r^4\over g(r)}-C_1^2r^2}}=d\phi$$
where $C_1$ is a positive constant.
It turned out that for $g(r)=r^2$, the equations of Geodesics has a general form
$$r=C_1 e^{C_2\phi}$$
where $C_2$ is another constant, this time, real.
I have two questions:
Q1
What is the general equation of geodesics for $g(r)={1\over r^2}+1$?
Q2
In case Q1 cannot be solved analytically for that $g(r)$ defined, for what kind of $g(r)$ could it be, having the following constraints on $g(r)$:
$$\lim_{r\to 0^+} g(r)=\infty$$ $$\lim_{r\to \infty} g(r)=1$$ ?