*I'm interested in finding the (unique?) solution to the set of delay differential equations*
$$f_w(w,x) = xf(w,w^2x)+w^3x^2f(w,w^4x), $$
$$f_x(w,x) = wf(w,w^2x)$$
With the initial condition $f(1,x) = e^x$ and $(w,x) \in \mathbb{C}^2$

For $|w| \leq 1$, I've found the series $f(w,x) = \sum_{n=0}^\infty \frac{w^{n^2} x^n}{n!}$ satisfies the equations, but it doesn't converge for $|w|>1$.

For $|w|>1$, I have considered the integral $$f(w,x)=\frac{1}{2 \sqrt{\pi}} \int_0^\infty \frac{e^{-\frac{\ln(t)^2}{4}}}{t} e^{x t^{\sqrt{\ln\left(w\right)}} }dt$$ which seems numerically to satisfy the differential equation, though it converges so badly at most places that its fairly useless.

**A few questions I have regarding this set of differential equations**

* Question 01*: Is the series solution unique? Is it the only smooth solution?

* Question 02*: How can we numerically approximate this delay differential equation?

*As far as I know, Mathematica's NDSolve doesn't support delay differential equations with non-constant delays, and it seems other computation programs similarly don't have support for this type of differential equation.*

* Question 03*: Are there methods to analytically approximate a solution?

*I'm considering that perhaps there is some way to resum the divergent series that gives an asymtotic solution*.