Why can I integrate two differentials on the two side of an equation?

Question

Suppose there is an ode: $$ \frac{dx}{dt} = a(t)x $$ Then we can solve it by following steps: $$ \frac{dx}{x} = a(t)dt $$ integrate on both side and we get: $$ \int \frac{dx}{x} = \int a(t)dt + C $$ Why can I integrate differentials, what does it exactly mean by integrate a differential? Is there an elementary explanation?

Answer 1

Suppose there is an equation of differentials: $$ F(x)dx = G(t)dt $$ The equation only makes sense when $x$ is exactly a function of $t$, we denote it by $x = x(t)$, then the equation becomes $$ F(x(t))dx(t) = G(t)dt $$ what the equation exactly says is that $$ F(x(t))x'(t)dt = G(t)dt $$ namely $$ F(x(t))x'(t) = G(t) $$ We can integrate on both side since they are functions and there is no ambiguity, and we get $$ \int F(x(t))x'(t)\, dt = \int G(t) \, dt + C $$ And $$ \int F(x(t))x'(t)\, dt = \int F(x) \, dx + C $$ is what we already know. So we get $$ \int F(x) \, dx = \int G(t) \, dt + C $$