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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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