I am interested in the following linear PDE:

$(1+ t\partial_t) ~ (\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x)A(x,t)+ {t\over (1-x)^3} A(x,t)=0$

with $A(x,0)=1$.

This PDE is invariant under the transformations $x \to {x\over x-1},~ t\to -t$, which fix the ratio $(1,6,6)$ in the x-derivative part.

This PDE looks like a Fuchsian-type equation, but (I think) normally Fuchsian equations assume only one variable – Does anyone know this type of "generalised Fuchsian-type equation" with two variables? I wonder if this is known and studied in the math literature.

I am interested in the PDE's analytic solutions, especially solutions that are non-perturbative in t, and other symmetry properties.

PS

For a slightly simpler PDE

$(1+ t\partial_t) ~ (\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x)A(x,t)+ t A(x,t)=0$

with $A(x,0)=0$

I found that the exact solution is given by a hypergeometric 0F3 function:

$A(x,t)=(x^2 t)~~{}_0F_3(~;{{4\over 3}, {5\over 3},2}; {-x^3 t\over 27})$.

This I believe corresponds to the small $x$ limit of the solution to the first PDE with the factor ${1\over (1-x)^3}$ included. I wonder if there is a way to generate the full exact solution to the first PDE.