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How can it be proved that $\pi$ is an irrational number through differential equations?

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    Well . . . . . . . . . .2012-11-13

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Hmm: Niven's proof of the irrationality of $\pi$ http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183510788 contains a differential equation $F''(x) + F(x) = f(x)$, although I don't know if you'd count it as a proof "through differential equations".