Do the equations of motion have specific characteristics? I solved a classical mechanics problem in a form somewhat like this:
$$x(t)=t^2+5t$$
$$y(t)=t^3$$
$$z(t)=5.$$
The problem asked me to find the equations of motion of an object. 
From my understanding, the "equations of motion" are simply equations that allow you to know the body's position, velocity, and acceleration. I thought my answer above met those conditions, yet it was marked wrong. Do the "equations of motion" in general have a specific form?
 A: You have written - presumably - a solution to the equations of motion, that is,
$$\vec r (t) = \begin{pmatrix}
t^2+5t\\ 
t^3\\ 
5
\end{pmatrix}$$
describing the position of a particle at any instant of time, $t\geq 0$, satisfying some initial conditions, in fact two, presuming the equations of motion are second order. You can find the velocity, or acceleration since,
$$\vec v(t) = \frac{d}{dt}\vec r(t) = \begin{pmatrix}
2t+5\\ 
3t^2\\ 
0
\end{pmatrix}, \quad \vec a (t) = \frac{d}{dt}\vec v (t) = \begin{pmatrix}
2\\ 
6t\\ 
0
\end{pmatrix}.$$
The equations of motion are differential equations to be solved for $\vec r(t)$, and whilst typically result from a Lagrangian and are at most second order in many cases, they need not be. An example of a set of equations of motion for your solution could be,
$$\left\{\begin{matrix}
\ddot x = 2\\ 
\ddot y = 6t\\ 
\ddot z = 0
\end{matrix}\right.$$
subject to $x(0) = y(0) = 0$, $z(0) = 5$ and $\dot x (0) = 5$, $\dot y (0) = \dot z(0) = 0$ which corresponds to starting the particle at position $(0,0,5)$ with a velocity of $5$ initially in the $x$-direction.
Note however this is not the only system of differential equations which may admit this solution, so without further information to narrow down choices, the question of finding 'the' equations of motion cannot be answered.

As for a general form of equations of motion, if the system is described by a Lagrangian $L$, then by definition it possesses equations of motion of the form,
$$\frac{\partial L}{\partial x} = - \frac{\partial}{\partial t} \frac{\partial L}{\partial \dot x}$$
as well as with respect to the other variables. For example, for a system described in polar coordinates, we would have two associated Euler-Lagrange equations w.r.t. $r$ and $\theta$.
A: An equation of motion describes how a mechanical system changes within space and time. Solving this equation will usually yield a function that will describe the state of the system at a given time $t$. Usually these are differential equations of second order, as the equation has to take in all the forces that act on the system (or that we don't ignore at least). Taking a standard example, a force acting on an mass particle $m$ will change its momentum. For $n$ forces acting on the system we obtain
$$\frac{{d\vec{p}}}{{dt}} = \sum_i \vec{F}_i$$
These forces cause a net acceleration $\vec{a}$ on the system, thus this is usually represented as:
$$\frac{{d\vec{p}}}{{d\vec{t}}} = m\vec{a}$$
Now let us go through some cases of these equations of motions.
We know that $$a=\frac{d^2\vec{r}}{dt^2}$$
In case no forces act on the object or they are all in equilibrium the object will not be accelerated:
$$\sum F = m\frac{d^2\vec{r}}{dt^2} = 0$$
Solving this equation yields the equation of motion for such a system:
$$\vec{r}(t) = \vec{v}_0 t + \vec{r}_0$$
so that an object without an initial velocity $\vec{v}_{0}$ and no net acceleration acting on it will not move.
In case the acceleration is constant solving the differential equation yields the famous equation:$$\vec{s}(t) = \frac{1}{2}\vec{a}t^2 + \vec{v}_{0}t + \vec{s}_0.$$
Substitute the acceleration with the earths gravitational acceleration and you get the term for the motion of an object in earths gravitational field (an approximation with limited applicability really).
Another famous equation of motion is for a harmonic oscillator: $$m\ddot{x} = -kx$$ which yields $$x(t) = x_{0} \sin(wt+\varphi_{0}).$$
Now compare these equations with each other. They all the describe the position of the system at a given time $t$. Of course this can get much more complicated, incorporating all three dimensions of space and also time: $$s = s(x,y,z,t)$$
And the acceleration may not be constant but dependent on many terms. This will make it much more complicated and might not allow for an analytic solution of the equation. You may also get chaotic motion or systems that depend strongly on initial conditions. 
In case of your equations you are dealing with a specific set of equations that describe a system, describing the path of the objects in all space dimensions at a given time $t$. So all equations of motions have something in common, but they are all special in their respective ways. Hope this clarified your question.
A: The above equations are consistent with a description of a specific continuous motion and so the position of the object requires no magic. 
There is no need to involve gravity or any central force.
May be the case that the equations do not describe the motion described in that particular problem.  
EDIT ADD:
after 2 nonsensical, uncommented downvotes, 
May the esteemed downvoters understand a graph, or maybe post a few words to make their point understandable.  
in sage 
the code 
t = var('t')
parametric_plot3d([t^2+5*t, t^3,5], (t,-3,3))  
will show the 3D trajectory from time -3 to 3 

