Consider the differential equation (frac{d^{2} y}{d x^{2}}-x y=0). Using the transformations (y=u sqrt{x}) and (frac{2}{3} i x^{3

Question:

Consider the differential equation $\frac{d^{2} y}{d x^{2}}-x y=0$. Using the transformations $y=u \sqrt{x}$ and $\frac{2}{3} i x^{3 / 2}=z$, where $i=\sqrt{-1}$, show that this differential equation reduces to the Bessel differential equation

\[
\frac{d^{2} u}{d z^{2}}+\frac{1}{z} \frac{d u}{d z}+u\left(1-\frac{1}{(3 z)^{2}}ight)=0
\]

What is the solution for $u(z)$ and $y(x)$ ?

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