possible*]{}, $$\lim_{r \rightarrow \infty}G_i(r,x) = 0$$ $$G_i(1,x) = 0$$ $$\frac{\partial\ }{\partial r}G_i(r,x)|_{r=1} = 0$$ However, there are significant complications. The Helmholtz term we have thought of as an modelling artifact has first to be understood as an optimal linear approximation to the full non-linear system. The first steps on this are considered in Section 8. We will eventually have to manage the fact that no $G_1$ function exists satisfying these boundary conditions. However, it is now consistent to proceed to construct $G_2$, as this (a) exists, and (b) is needed for the management of the full non-linear system and its solution.

Integration of the inertia term
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We will now focus on the lowest order correction arising from the inertia term, for which $G_2$ is the required radial Green’s function. The building blocks for this are two different solutions of the ODE $$\biggl(\frac{d^2 \ }{d r^2} + \frac{1}{r} \frac{d\ }{dr} - \frac{2^2}{r^2}\biggr)^2 \phi(r) = 0\ .$$ The solution to the homogeneous problem is of the form $$\phi(r) = a r^4 + b r^2 + c + \frac{d}{r^2}$$ To try to build a Green’s function, we would write down a pair of solutions: $$\begin{split}
\phi_L(r) &= a_L r^4 + b_L r^2 