So I have the following integral: $$Z(a,b,c) = \int dx dy e^{-\frac{1}{\hbar}(x^2 + y^2 +ax +by +cx^4 +dy^4)}$$
I want to obtain the Ward identity so I take $$0= \int dx dy \frac{d}{dx} \frac{d}{dy} \Big( exp[-\frac{1}{\hbar}(x^2 + y^2 +ax +by +cx^4 +dy^4)] \Big) = (\frac{-1}{\hbar})^2 (2y + b + 4y^3c)(2x+a+4cx^3) Z(a,b,c).$$
Now, expressing this in terms of derivatives of a, b, and c I get:
$$(2\frac{d}{db} + c\frac{d}{dc})(2 \frac{d}{da} + a + c \frac{d}{dc}) Z(a,b,c)=0$$
which I think is the Ward identity.
Now, this Ward identity is supposed to tell me something about the integral but I don't know what that would be. The only observation I can make is that there are an infinite set of solutions to the above differential equation, this also infinitely many Ward identities, I think? But again that don't tell me much about the integral?