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How do I show that the function $$ y(x) = \int_0^\infty \cos(t^3/3 + xt)~dt\qquad -\infty \lt x \lt \infty,$$ satisfies the differential equation $y''=xy$?

I can't simply differentiate under integral since the integrand is oscillatory and does not decay as $t$ becomes large.

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    Ah, Airy's differential equation. Have you tried maybe a preliminary substitution, or integration by parts on your integral? Alternatively: you've tagged [tag:complex-analysis], so you could maybe try using a different integration path...2012-04-17
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    Does this integral even converge? I'm not answering OP's question at all by asking this, but I'm always too much suspicious when it comes to differential equation... and this has me worried.2012-04-20

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