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enter image description here I was struggling to understand this footnote from the book by Gelfand. I couldnot understand how he takes the solution of the differential equation enter image description here outside the interval $[a,b]$ and also how will the solution of $(26)$ with initial condition $$h(a-\epsilon)=0$$ $$h'(a-\epsilon)=1$$ Cannot vanish on whole interval $[a,b]$

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  • $\begingroup$ A conjugate point is a second/the next greater root of $h$? $\endgroup$ Nov 6 '19 at 13:20
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$P,Q$ are defined outside $[a,b]$ by the implicit assumptions of the cited footmark. Also, if $P$ is non-zero on $[a,b]$, then it is also non-zero on some open neighborhood of that interval. A linear ODE has solutions that can be extended to the full intervals over which the coefficients are continuous and the leading coefficient is non-zero.

This all makes it possible to take the initial condition at $a−ϵ$ for a sufficiently small positive $ϵ$.

For the overall claim, the conjugate point is a continuous function of the initial point. If it is greater than $b$ for initial point $a$, then that same is true for initial points in a neighborhood of $a$, simply by continuity.

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  • $\begingroup$ "the conjugate point is a continuous function of the initial point", i dont know about this can you shed some light about it. $\endgroup$
    – Uncool
    Nov 7 '19 at 8:57
  • $\begingroup$ This is the meaning of the first sentence: "The solution ... depends continuously on the initial conditions" and by the implicit function theorem, the root depends continuously on the solution, as it has to be a simple one (as all roots of a non-trivial solution of a 2nd order linear DE have to be). $\endgroup$ Nov 7 '19 at 9:35

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