Attractive central force problem with impact parameter

Let $m$ be a mass subjected to a central force:

$$vec{F}(r) = -frac{k mu}{r^3} vec{u_r} $$

Where $k$ is a positive constant and $mu$ is the reduced mass of the system with the centre of forces situated at the origin of coordinates $O$. Its orbit equation in terms of the angle $phi$ is ($u = 1/r$):

$$u”(phi) + u(phi)(1-frac{kmu^2}{l^2}) = 0$$

Where $l$ is the angular momentum. Then I have to solve this equation for a specific case where the initial conditions are the following: the particle is at an infinite distance from the origin of the center of force $O$, it moves anti-parallel to the axis $X$ with impact parameter $d$ with respect to $O$, and the modulus of its velocity is $v$ (see figure).

Attempt: I have solved the differential equation for two cases:

1. The term $1-frac{kmu^2}{l^2}$ is positive, so $k < l^2/mu^2$ and the general solution is:

$$u(phi) = Asin({sqrt{1-frac{kmu^2}{l^2}}phi + delta})$$

Does this mean the particle will do an SHM in one dimension passing through the centre of force?

2. The term $1-frac{kmu^2}{l^2}$ is negative, so $k > l^2/mu^2$ and the general solution is:

$$u(phi) = Ae^{sqrt{1-frac{kmu^2}{l^2}}phi} + Be^{-sqrt{1-frac{kmu^2}{l^2}}phi}$$

Where depending on the $A$ and $B$ values, the movement will spiral around the centre (bounded) (1) or from the centre to infinity (unbounded) (2).

However, I cannot identify which one of these cases fit with the problem. I guess is the first one of the second case, but I’m not pretty sure, because I haven’t done any exercise about attractive forces with an impact parameter. Could anybody help me?

Thanks in advance!