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Solution method

In the quotient M (t x, t y) N (t x, t y) = M (x, y) N (x, y) {\textstyle {\frac {M(tx,ty)}{N(tx,ty)}}={\frac {M(x,y)}{N(x,y)}}}, we can let t = 1 / x to simplify this quotient to a function f of the single variable y / x:

That is

Introduce the change of variables y = ux; differentiate using the product rule:

This transforms the original differential equation into the separable form

or

which can now be integrated directly: ln x equals the antiderivative of the right-hand side (see ordinary differential equation).