I have obtained the correct solution to this PDE but my method required me to treat the partial dv/dsigma as if it were a fraction which im fairly sure is invalid. Could someone confirm this for me and guide me towards a correct method?

I have obtained the correct solution to this PDE but my method required me to treat the partial dv/dsigma as if it were a fraction which im fairly sure is invalid. Could someone confirm this for me and guide me towards a correct method?

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