(I'm sorry for the format of the mathematical symbols in advance)
Take the differential equation dy/dx = x * y, for example, the process for solving it goes as follows.
1.) $dy = x * y * dx$
2.) $1/y * dy = x * dx$
3.) $∫ 1/y dy = ∫ x dx$
4.) $ln(y) = (x^2)/2$
5.) e^ln(y) = e^((x^2)/2)
6.) y = e^((x^2)/2)
Now my question is in regards to step 3. When we take the integral of both sides why do we not treat the infinitesimals dy and dx as part of the function that we are integrating? In other words, when taking the integrals why didn't we get $∫ 1/y dy dx = ∫ x dx dx$ instead? What happened to them?
P.S. I am currently taking the regular high school Calculus 2 course so please keep the "advanced" mathematics to a bare minimum when answering the question.
$$ y'= xy $$ $$ \frac {y'}{y}= x $$ Take inetgral now on both sides
$$ \color{red}{\int} \frac {y'}{y} \color{red}{dx}=\color{red}{\int} x \color{red}{dx}$$ $$ \color{red}{\int} \frac 1y \frac {dy}{dx}\color{red}{dx}=\color{red}{\int} x \color{red}{dx}$$ $$\int \frac {dy}{y}=\int x dx$$
I prefer this method $$\frac {y'}{y}= x$$ $$(\ln(y))'= x$$ Then i take integral on both sides $$\color{red}{\int }(\ln(y))'\color{red}{dx}= \color{red}\int x \color{red}{dx}$$ $$\int \frac {d(\ln(y))}{dx}dx= \int x dx$$ $$\int {d(\ln(y))}= \int x dx$$ $$\ln(y)= \frac {x^2}2 +K$$