The curve is usually called a trajectory or an orbit. The (vector) differential equation is reformulated as a geometrical description of the curve, that is, as a differential equation in terms of the phase space variables only, without the original time parametrization.
The curve is commonly referred to as a path or an orbit. The (vector) differential equation is rephrased into a geometric depiction of the curve, which means expressing it in terms of phase space variables alone without using the original time parameterization.