Show that $\int_{0}^{\pi} \mathrm{e}^{i x \cos t} d t$ satisfies the differential equation

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

and find a second solution, in the form of an integral, for $x>0$.

Show, by finding the asymptotic behaviour as $x \rightarrow+\infty$, that your two solutions are linearly independent.