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For example, given the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}x}=y\tag*{}$ We do, $\dfrac{\mathrm{d}y}{y}=\mathrm{d}x\tag{*}$ At which point we go and integrate both sides. Is there a rigorous justification for being able to go to $(*)?$

Archimedesprinciple
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    Also https://math.stackexchange.com/questions/1252405/is-it-mathematically-valid-to-separate-variables-in-a-differential-equation –  Jun 17 '17 at 12:30
  • this is only possible if $$y\ne 0$$ – Dr. Sonnhard Graubner Jun 17 '17 at 13:57
  • There are so many related posts on this particular question: https://math.stackexchange.com/questions/1906241/when-not-to-treat-dy-dx-as-a-fraction-in-single-variable-calculus/. – StubbornAtom Jun 17 '17 at 18:08

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Basically you are using the fundamental theorem of calculus and the chain rule (combined together they constitute the rule of integration by substitution). When you write $f(y) dy = g(x) dx$ you are really writing $f(y(x)) \frac{dy}{dx} = g(x)$. You then integrate both sides over the same limits, obtaining

$$\int_a^b f(y(x)) \frac{dy}{dx} dx = \int_a^b g(x) dx$$

The left side is

$$\int_{y(a)}^{y(b)} f(y) dy$$

so that "dropping the integral signs" gives

$$f(y) dy = g(x) dx.$$

But in the standard framework this last equation does not really mean anything.

Ian
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