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This is the Airy equation, and its two independent solutions are $Ai(-x)$ and $Bi(-x)$. Both of which oscillate with a decaying amplitude but an increasing frequency as $x\rightarrow \infty $.

That means a general solution has also similar behavior.

Can anyone give a direct proof that a general solution of the equation does not diverge as $x\rightarrow \infty$?

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  • $\begingroup$ Have you tried using power series methods? $\endgroup$ – Dr. Ikjyot Singh Kohli Dec 29 '17 at 4:42
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    $\begingroup$ The standard approach is to represent the solution to the equation as an integral transform then apply the method of steepest descents to obtain an approximation. See section 2.2.1 of these notes. $\endgroup$ – Antonio Vargas Dec 29 '17 at 19:00

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