fundamental set of solutions
A homogeneous linear ODE has not one solution but a whole space of them, and like any vector space that space has a basis — a smallest collection from which every member can be built by combination. A fundamental set of solutions is exactly that basis. Find it once and you have, in effect, found every solution at the same time.
Precisely, for an n-th order linear homogeneous ODE on an interval, a fundamental set is any collection of n linearly independent solutions y_1, ..., y_n (equivalently, n solutions whose Wronskian is nonzero). The existence-uniqueness theorem guarantees such a set exists. Once you have it, the general solution is y = c_1 y_1 + ... + c_n y_n, and the n constants can be chosen to match any prescribed initial conditions — that is what makes the set complete, not merely independent.
The fundamental set is the goal of the first half of every linear-ODE problem: from the characteristic equation you read off n independent solutions (exponentials, polynomials times exponentials, sines and cosines), assemble them into the homogeneous solution, then add a particular solution. The same idea generalizes: in Sturm-Liouville theory the eigenfunctions play the role of a fundamental set for expansions, and for systems the columns of the fundamental matrix are the vector analog.
For y'' - 3y' + 2y = 0 the characteristic roots are 1 and 2, so { e^x, e^(2x) } is a fundamental set, and the general solution is y = c_1 e^x + c_2 e^(2x).
Two independent solutions span all solutions of a second-order homogeneous linear ODE.
You need exactly n solutions for order n — no more, no fewer. n+1 'solutions' must be dependent; fewer than n cannot meet all initial conditions. A fundamental set is not unique: any independent re-combination is another valid set.