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[tex]\left (\frac{\mathrm{d} y}{\mathrm{d} x} \right )^n + \left (\frac{\mathrm{d} y}{\mathrm{d} x} \right )^{n-1} + \left (\frac{\mathrm{d} y}{\mathrm{d} x} \right )^{n-3}... = 0[/tex]

Has some ancient greek guy managed to give a name and techniques on how to solve this?

Do ODEs become nearly impossible if I throw in a g(x,y) or h(x,y) in there? That is

[tex]H(x,y)\left (\frac{\mathrm{d} y}{\mathrm{d} x} \right )^n + G(x,y)\left (\frac{\mathrm{d} y}{\mathrm{d} x} \right )^{n-1} + \left (\frac{\mathrm{d} y}{\mathrm{d} x} \right )^{n-3}... = 0[/tex]

I imagine it would because we don't even have a "clean" formula for solving cubics.

Is there a method if it was quadratic?

How bad do the G(x,y) and H(x,y) messes things?