Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 31E (Asymptotic Methods)

After division by $x^2$, the equation becomesThe regular singular point criterion for a second-order equation would require $x(1/x)$ and $x^2(-1/x^4)$ to be analytic at zero. The second expression is $-x^{-2}$, so the criterion fails:

The removal of the first derivative from a second-order differential equation usesIt givesso

Write the normal-form equation asFor sufficiently small positive $x$, $Q(x)>0$. Moreover,and the corresponding higher derivative condition also tends to zero, so the amplitude varies slowly relative to the exponential phase and the Liouville-Green approximation applies.

The phase and amplitude have expansionsHence two independent Liouville-Green solutions satisfyIn particular, after choosing the signs according to growth and decay,

Since $y=x^{-1/2}u$, the two linearly independent solutions of the original equation have the asymptotic expansionsAs a check, the substitution $z=1/x$ converts the equation into the modified Bessel equation of order zero, whose solutions $I_0(z)$ and $K_0(z)$ have exactly these large-$z$ expansions.