I have a two-variable recurrence relation which was given to me, and I've been playing around with it. I am familiar with the method of generating functions, but not so much with differential equations. This was one representation that I derived, and while I believe I have something else in one variable that may be more promising, I was wondering if this was a known form:
$$Q(x,y) = x^2\left[\frac{\partial}{\partial x} Q(x,y)\right] +xy \left[\frac{\partial}{\partial y}Q(x,y)\right] + x^2y\left[\frac{\partial}{\partial x} Q(x,y)\right]$$
I have no experience in PDE's, so I have no grounds for anticipating this to be intractable v. well-known.
Thanks!
You can use the Method of Characteristics - Lagrange-Charpit Equations. You need to solve these differential equations:
$$\dfrac {dx}{x^2(1+y)}=\dfrac {dy}{xy} =\dfrac {dQ}{1}$$ The first differential equation is separable: $$\dfrac {dx}{x^2(1+y)}=\dfrac {dy}{xy} $$ $$\int \dfrac {dx}{x}=\int \dfrac {1+y}{y} dy $$ $$\ln x= \ln y +y +k$$ $$x=C_1ye^y \implies C_1=\dfrac {x}{ye^y}$$ Now solve the second DE: $$\dfrac {dy}{xy} =\dfrac {dQ}{1}$$ $$\int \dfrac {dy}{y^2e^y} =C_1Q$$ This is not easy to integrate.