Asymptotic solutions to ODE Wondering where I can find resources to do more questions of the following type and also if you guys can help me answer this problem.

Consider the differential equation:
$$u'' + \left( 1-\frac{\gamma}{x^2} \right) u = 0$$
for $x > 4$. Obtain the first two terms of the asymptotic solution for each of the two real solutions of this equation.

We start by writing out $u = u_0+u_1$ and consider the zeroth order solution which is $$u_0''+u_0=0$$ This has solution $u_0=e^{ix}$. Basically how do I proceed from here? Also where can I find more problems requiring this method of solution. Thanks!

An expansion of $u $ derived from the perturbation theory would be $u=u_0 +\gamma u_1 + \dots$ The $0$th-order and $1$st-order equations (in terms of powers of $\gamma$) are respectively
\begin{aligned}
u''_0 + u_0 &= 0 \, ,\\
u''_1 + u_1 &= \frac{u_0}{x^2} \, .
\end{aligned}
The $0$th-order solution is $$u_0(x) = a_0\cos x + b_0\sin x \, ,$$
whereas there is more work to find the $1$st-order solution. Indeed, $$u_1(x) = a_1\cos x + b_1\sin x + u_p(x)\, ,$$
where $u_p$ is a particular solution to the non-homogeneous ODE satisfied by $u_1$. The initial conditions $u(4) = U$, $u'(4) = V$ are first applied on $u_0$, $u'_0$:
\begin{aligned}
a_0\cos 4 + b_0\sin 4 &= U \, ,\\
b_0\cos 4 - a_0\sin 4 &= V \, ,
\end{aligned}
which yields $a_0 = U\cos 4 - V\sin 4$ and $b_0 = U\sin 4 + V\cos 4$.
Then, they are applied on $u_0 +\gamma u_1$, $u'_0 + \gamma u'_1$, etc.
There is much more to read about such methods in the book Perturbation Methods by A.H. Nayfeh (Wiley, 2008).

This linear ODE may be solved analytically. The general solution involves Bessel functions.
