Can the Inverse Operator be used to solve PDEs?

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SUMMARY

The discussion centers on the application of the inverse operator, specifically \(\frac{1}{D}\), where \(D\) represents the derivative operator, to solve partial differential equations (PDEs). The user proposes a generalization to two dimensions with \(\frac{1}{D_{x}+D_{y}}\) and explores its implications through the equation \(\frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}=f\). The challenge remains in finding a solution for \(f\) from the derived equation, indicating a gap in the current understanding of inverse operators in the context of PDEs.

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

[tex]\frac{1}{D}f= \int dx f(x)[/tex]

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

[tex]\frac{1}{D_{x}+D_{y}}f= \iint dxdy f(x,y)[/tex]

(Double indefinite integral over x and y) and so on ...
 
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Did you even try it on a few examples?
 
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I have tried this:

[tex]\frac{1}{D_{x}+D_{y})f=g[/tex]

then we can put it in the form:

[tex]\frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}=f[/tex]

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

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