Solution Branching in Linear Differential Equations

Given a nontrivial y, solution branches of the form z ≡ y + ∑mi = 1ciwi are constructed with wi = 1, 1 ≤ i ≤ m. Multiple zeros of y "unwind" to nearby simple zeros of the branch z. Key assumptions are 1-dimensionality of the solution space of the boundary value problem, an exactness condition on the zeros of y, and a restriction on multiple zero counts. The branching lemmas and theorems are applied to the theory of conjugate points and extremal solutions, and also to branching of solutions of y(n) + p(x) y = 0. If l-dimensionality is dropped, then solution branches z exist but the constants ci, may not be small.