# Differential Equation dependent or independent

I am asked to state whether the variable is independent or dependent for the equation $$\frac{d}{dx}\left(x\frac{df}{dx}\right)+f=0$$ How do you know which are dependent and independent? What about for the equation: $$\frac{dy}{dx}=\frac{1}{\log \left| x+x^2+e^x+e^{x^2} \right|}?$$

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In my humble opinion, these questions are ridiculous. I do not think that selecting the unknown is the major task for people who are learning ODEs. Anyway, a reasonable test is to check which variable is in the "denominator" of the derivative. Since you have $d/dx$, you should conclude that $x$ is the independent variable. If you have only one other variable, it will be the dependent one. If you have more, it will be in the "numerator" of the derivative.
Reminds me of the joke: the function $e^x$ walks down the street, and sees the denizens fleeing toward him in fear, screaming: "the derivative is coming, the derivative is coming!" $e^x$ turns to look, gives a small chuckle and calls out to them: "have no fear, for I am immune to differentiation." At that moment the derivative taps $e^x$ on the shoulder and introduces himself, "Hi, I am $\tfrac{d}{dy}$." – Willie Wong Sep 24 '12 at 12:10