- #1

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## Main Question or Discussion Point

Hey All,

Can someone please give me the gist of how to show that the integral form of the Airy function for real inputs:

[tex]

Ai(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\tfrac13t^3 + xt\right)\, dt,

[/tex]

satisfies the Airy Differential Equation: y'' - xy = 0

I tried differentiating twice wrt to the x variable (assuming I could just bring it inside the integration) and then subbing back into the ODE but that failed.

Regards,

Thrillhouse

Can someone please give me the gist of how to show that the integral form of the Airy function for real inputs:

[tex]

Ai(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\tfrac13t^3 + xt\right)\, dt,

[/tex]

satisfies the Airy Differential Equation: y'' - xy = 0

I tried differentiating twice wrt to the x variable (assuming I could just bring it inside the integration) and then subbing back into the ODE but that failed.

Regards,

Thrillhouse