Chapter 12.23, Problem 18MP

(a) Make the change of variables $z=e^x$ in the differential equation $y^{\prime \prime }+e^{2x}y=0$ and so find a solution of the differential equation in terms of Bessel functions.

(b) Make the change of variables $z=e^{x^2/2}$ in the differential equation $x{y}^{\prime \prime }-y^{\prime }+$

$x^3\left(e^{x^2}-p^2\right)y=0,$ and solve the equation in terms of Bessel functions.