Can the Inverse Operator be used to solve PDEs?

[tpm]

Don't know if I'm wrong or this makes sense but if:

$$\frac{1}{D}f= \int dx f(x)$$

Where D is the derivative operator and the integral is indefinite, my question is if as a generalization of this then:

$$\frac{1}{D_{x}+D_{y}}f= \iint dxdy f(x,y)$$

(Double indefinite integral over x and y) and so on ...

 

[matt grime]

Did you even try it on a few examples?

 

[tpm]

I have tried this:

$$\frac{1}{D_{x}+D_{y})f=g$$

then we can put it in the form:

$$\frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}=f$$

HOwever i don't know how to solve this PDE to get 'f'