I could solve it by substitution if the exponent of $\log y$ was $y$ instead of $x$ but I've no idea how to solve this one. Is there some method to solve it? I don't think it can be solved in terms of standard functions.
Solution of the differential equation $\frac{dx}{x(\log x)^{x}}=\frac{dy}{y(\log y)^{x}}$
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ordinary-differential-equations