If $Z=\frac{1}{\partial_{x}^{2}+\partial_{x}\partial_{y}-12\partial_{y}^{2}}.y e^{3x+y}$, what is $Z$?

Question

If: $$Z=\frac{1}{\partial_{x}^{2}+\partial_{x}\partial_{y}-12\partial_{y}^{2}}y e^{3x+y}$$

Then what is $Z$?

I know that:

$$\frac{1}{\partial_{x}^{2}+\partial_{x}\partial_{y}-12\partial_{y}^{2}}y e^{3x+y}=e^{3x+y}\frac{1}{(\partial_{x}+3)^{2}+(\partial_{x}+3)(\partial_{y}+1)-12(\partial_{y}+1)^{2}}y $$

but I don't know the value of: $$\frac{1}{(\partial_{x}+3)^{2}+(\partial_{x}+3)(\partial_{y}+1)-12(\partial_{y}+1)^{2}}y$$

Is there a general way to find the value of equations of the form: $$\frac{1}{\phi(\partial_{x},\partial_{y})}f(x,y)?$$

The answer is actually $Z=e^{3x+y}{\frac{1}{98 }(21 x^3 - 8 x + 14 x y)}$

Answer 1

By writing $Z = e^{3x+y}W$, we have

$$ \{ (\partial_x + 3)^2 + (\partial_x + 3)(\partial_y + 1) - 12(\partial_y + 1)^2 \} W = y. $$

If we make an ansatz that $W = W(y)$ depends only on $y$, we can simplify the equation by

$$ -12 W''(y) - 21W'(y) = y \quad \Longleftrightarrow \quad W(y) =-\frac{y^2}{42} + \frac{4y}{147} + c_1 + c_2 e^{-7y/4}. $$

This gives one family of solutions. I am not sure whether there are other families of solutions.

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p.s. This kind of equation in general has no unique solution, which means that you can have a variety of solutions. Indeed, Mathematica shows that

You can try this code as follows:

L[z_] := D[z, {x, 2}] + D[D[z, x], y] - 12 D[z, {y, 2}];
L[Exp[3 x + y] (-(y^2/42) + (4 y)/147 + C[1] + C[2] Exp[-7 y/4])] // Simplify
L[Exp[3 x + y] (21/98 x^2 - 8/98 x + 14/98 x y)] // Simplify