To the end of the question, you have worked your way towards a more realistic picture, the initial premise is formulated unfortunately, since every differential equation has an infinity of solutions.
Look at the problem from a different perspective, the one of the existence and uniqueness theorem. It states, more or less explicitly, that there is a one-on-one relation between solutions and initial conditions. Since the initial conditions are from an $n$-dimensional space, the solution manifold inherits this dimension, trivially via the map to the values at the initial point.
See also the theory around the flow of a differential equation.