Solving separable differential equations by formal computations

Question

When solving a separable differential equation we move from the original equation $$N(y)\dfrac{dy}{dx}=M(x)$$ to $$N(y)dy=M(x)dx$$ Then we integrate both sides. My question is how precise is the expression $N(y)dy=M(x)dx$ ? is it a formal writing to simplify computation and get quickly into integrating both sides or is it a precise mathematical expression that has a precise mathematical meaning but goes beyond an introductory course on differential equations? Thanks for your help !

Answer 1

It is more like integrating the first equation with respect to $x$ on both sides. And then, taking $y$ is a function of $x$ which it is as they both are related by this equation, let say

$y = f(x)$ so,

$dy = f^{'}(x)dx$ or $\frac{dy}{dx} = f^{'}(x)$

Replacing this in the first equation we have

$N(y)f^{'}(x)dx$ = $N(y)dy = M(x)dx$

as $dy = f^{'}(x)dx$