Show that

$$I_{0}(x)=\frac{1}{\pi} \int_{0}^{\pi} e^{x \cos \theta} d \theta$$

is a solution to the equation

$$x y^{\prime \prime}+y^{\prime}-x y=0,$$

and obtain the first two terms in the asymptotic expansion of $I_{0}(x)$ as $x \rightarrow+\infty$.

For $x>0$, define a new dependent variable $w(x)=x^{\frac{1}{2}} y(x)$, and show that if $y$ solves the preceding equation then

$$w^{\prime \prime}+\left(\frac{1}{4 x^{2}}-1\right) w=0 .$$

Obtain the Liouville-Green approximate solutions to this equation for large positive $x$, and compare with your asymptotic expansion for $I_{0}(x)$ at the leading order.