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Given that $w'' = zw$, and $w(0) =1$. Prove that $w(x) = \int_o^\infty \cos(t^{3} - xt) \; dt$ satisfies $w'' = \frac{1}{3} xw$. Also, evaluate $w(0)$ and $w'(0)$ in the following: $\int_0^\infty \cos(t^{3}) \; dt$= $\int_0^\infty \frac{\cos x}{x^\frac{2}{3}} \; dx$. I am lost on how to start this problem.

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Differentiate under the integral sign to begin with. – Hans Lundmark Apr 9 '12 at 6:35
What's $z$? How does $w''=zw$ fit with $w''=\frac13xw$? – joriki Jul 24 '12 at 6:53

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