Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without directly solving the differential equation?
For example, consider the differential equation \begin{gather} \frac{dz}{d\phi} = \lambda e^{i \phi} z - 1 \end{gather} where $\lambda$ and $\phi$ are real, but $z$ is complex. This has solution \begin{gather} z_A(\phi) = A e^{- i \lambda \exp(i \phi)} + i e^{- i \lambda \exp(i \phi)} \operatorname{Ei} \left( i \lambda e^{i \phi} \right) \end{gather} where $A$ is a constant and $\operatorname{Ei}$ is the exponential integral. I therefore conclude that the general solution for $z$ in the above differential equation has Stokes lines that correspond to the Stokes lines for $\operatorname{Ei}(i \lambda e^{i \phi})$, viz., $\phi = \pi / 2$ and $\phi = - \pi / 2$. This is all very well, but consider that I now generalise the above differential equation in some way, e.g., write $\lambda = \lambda(z)$, add a term proportional to $z^2$, or do something even more exotic, such that I can no longer find a solution. Can one nevertheless determine the Stokes lines and anti-Stokes lines that a general solution has?