# Article

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Keywords:
eventual disconjugacy
Summary:
The work characterizes when is the equation $y^{ (n) } + \mu p(x) y = 0$ eventually disconjugate for every value of $\mu$ and gives an explicit necessary and sufficient integral criterion for it. For suitable integers $q$, the eventually disconjugate (and disfocal) equation has 2-dimensional subspaces of solutions $y$ such that $y^{ (i) } > 0$, $i = 0, \ldots , q-1$, $(-1)^{i-q} y^{ (i) } > 0$, $i = q, \ldots , n$. We characterize the “smallest” of such solutions and conjecture the shape of the “largest” one. Examples demonstrate that the estimates are sharp.
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