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 Mentor P: 4,499 Separation of variables can be explained by the chain rule. If your differential equation is $$\frac{dy}{dx} = f(x)/g(y)$$ (in your example, f(x)=k and g(y)=1/y) then we can write $$g(y) \frac{dy}{dx} dx = f(x)$$ These are both functions of x (because g(y) = g(y(x))). So integrate them both $$\int g(y) \frac{dy}{dx} dx = \int f(x)dx$$ and the result follows from noting that $$\int g(y(x)) \frac{dy}{dx} dx = \int g(y)dy$$ by doing integration by parts (with the substitution y=y(x)). Then you can consider the act of separating the variables as just a mnemonic shortcut to avoid a lot of notation