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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.

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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
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