Exponential estimates for solutions of 𝑦^{”}-𝑞²𝑦=0

Read, T. T.

doi:10.1090/S0002-9939-1974-0344611-5

Exponential estimates for solutions of $y^{”}-q^{2}y=0$
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by T. T. Read PDF

Proc. Amer. Math. Soc. 45 (1974), 332-338 Request permission

Abstract:

It is shown for any nonnegative continuous function $q$ on $[0,\infty )$ and any $c < 1$ that any positive increasing solution $y$ of $y'' - {q^2}y = 0$ satisfies $y(x) \geq y(0)\exp (c\int _0^x {q(t)dt)}$ on the complement of a set of finite Lebesgue measure. It is also shown that if $\lim \inf (\int _0^x {q(t)dt/x) > 0}$ then the equation has an exponentially increasing solution and an exponentially decreasing solution.

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