See the bottom for an added definition and discussion.
Context
Note: I've never had a course in differential equations, so my question may demonstrate gross ignorance.When the distinction is clear, I like to denote dependent quantities differently than independent quantities. For example, $\mathfrak{r}$ represents the instantaneous value of a radial position vector; whereas $\vec{\mathit{r}}\left[t\right]$ is the analogous radial position vector function of time.
This is as much an art as a science, as in following listing of various quantities describing the "instantaneous" motion of a point-mass whose position $\mathfrak{r}$ is given as a function $\vec{\mathit{r}}\left[t\right]$ of time. There was no conscious intent to decouple velocity $\vec{\mathit{v}}\left[t\right]$ from position. It just seemed cleaner to write velocity as a function of time, rather than as a time derivative. But those are the only quantities which I am treating as functions, and only in so much as they are evaluated once during the exercise.
The Central Force Problem (CFP)
A typical statement of the central force problem is: find the solutions to the vector differential equation
\begin{align} m\ddot{\mathfrak{r}}=-mf\left(\mathfrak{r}\right)\frac{\mathfrak{r}}{r}, \end{align}
where $f$ is a known function of position. Mass $m$ is a constant.
So, clearly $f$ is a function (dependent variable). Since we have a second derivative of position with respect to time, it seems reasonable to treat position as a function of time. Since acceleration is given as a function of position, it might be reasonable to consider $f$ and $\ddot{\mathbf{r}}=\ddot{\vec{\mathit{r}}}\left(\mathfrak{r}\right)$ as functions of $\mathfrak{r}=\vec{\mathit{r}}\left(t\right).$ It may be a bit nebulous to ask, but; is this a good way to think about the generic CFP? That is, should the different terms be treated as dependent variables with the suggested order of dependency?
The Kepler Problem
Hopefully, the second part of my question is more tangible. There are multiple ways to arrive at Kepler's three laws if we replace $f$ with Newton's inverse square law. At least in the case where $\left\Vert\mathfrak{r}\right\Vert$ is either a maximum or a minimum, given an initial velocity and initial position, the shape and period of the obit is determined. We also have that the areal velocity (or equivalently, the angular momentum) is a constant with respect to time. What we don't have, in general, is a closed form expression for position as a function of time. The the cases of exponential growth and the harmonic oscillator, the solutions are explicit functions of time.
So I ask: are Kepler's laws a solution to the Kepler problem in the sense of being solutions to the differential equation
\begin{align} m\ddot{\mathfrak{r}}=-G\frac{mM\mathfrak{r}}{r^3}? \end{align}
Edit to add:
Solution Defined
This is from The Geometry of Celestial Mechanics, by Hansjörg Geiges.
By a solution of the (CFP) we mean a $\mathscr{C}^2$-map $\mathbf{r}:I\to\mathbb{R}^3\backslash\left\{\mathbf{0}\right\}$ defined on some interval $I\to\mathbb{R},$ that satisfies the differential equation (CFP).
I guess if I keep reading, I may find an answer to my question. At first sight, it looks to be a clear affirmation that Kepler's laws provide a solution to the Kepler problem. But what does this definition really mean? The mapping $\mathbf{r}$ implies a parameter. Should $\ddot{\mathbf{r}}$ mean a derivative with respect to that parameter? If so, then Kepler's ellipse isn't sufficient. If it merely means a parameterized set of points at which (CFP) is satisfied, and leaves such details as the meaning of $\ddot{\mathbf{r}}$ to context, then we apparently do have a solution.
My inclination is to say: No, the ellipse is not a complete solution. But I am not an authority on the topic.