the n-dimensional solution space
Think of all the solutions of a homogeneous linear equation as living together in one room. A wonderful fact is that this room is not a chaotic jumble — it is a flat, well-organized space, exactly like ordinary vectors. You can add two solutions and get a solution; you can scale a solution and get a solution. And for an nth-order equation, the room has exactly n dimensions: n independent directions, no more and no fewer.
Here is what that means in practice. Take the homogeneous equation y^(n) + p_(n-1)(x) y^(n-1) + ... + p_0(x) y = 0. By the existence-uniqueness theorem, a solution is pinned down by its value and its first n-1 derivatives at one point — that is n numbers of freedom. So the set of all solutions matches up perfectly with n-tuples of starting data, which is an n-dimensional space. Concretely, you can find n solutions y1, y2, ..., yn that are linearly independent (a fundamental set), and then EVERY solution is a unique combination y = c1 y1 + c2 y2 + ... + cn yn. The n constants are the n coordinates of a solution in this space.
This single number, n, is the organizing principle of the whole linear theory. It tells you how many independent solutions to hunt for before you can stop, why an nth-order initial value problem needs exactly n initial conditions, and why the general solution has exactly n arbitrary constants. The caveat: this clean dimension-n picture is guaranteed for the HOMOGENEOUS equation on an interval where the coefficients are continuous; the forced equation's solutions form a shifted copy of this space (an affine space), not a space through the origin.
For the fourth-order equation y^(4) - y = 0 (n = 4), four independent solutions are e^x, e^(-x), cos(x), sin(x). The solution space is four-dimensional, and the general solution y = c1 e^x + c2 e^(-x) + c3 cos(x) + c4 sin(x) names every solution by its four coordinates.
An order-n homogeneous equation has a solution space of dimension exactly n.
Dimension n is the count of INDEPENDENT solutions, not the number you happen to write down — five solutions of a third-order equation must include redundancy, because the space is only three-dimensional.