Determine the range of the integer $n$ for which the equation

$$\frac{d^{2} y}{d z^{2}}=z^{n} y$$

has an essential singularity at $z=\infty$.

Use the Liouville-Green method to find the leading asymptotic approximation to two independent solutions of

$$\frac{d^{2} y}{d z^{2}}=z^{3} y$$

for large $|z|$. Find the Stokes lines for these approximate solutions. For what range of $\arg z$ is the approximate solution which decays exponentially along the positive $z$-axis an asymptotic approximation to an exact solution with this exponential decay?