Integrating both sides of an equation with respect to what? Let's say we have the following DE:
$ \frac{dy}{dx} = x $
1. That's how It could be solved:
$dy = x dx$
$\int{dy}=\int{xdx}$
$y=\frac{1}{2}x^2+c$
Is it mathematically correct to separate $dy$ and $dx$ ? Or it would appear as the derivative of $y$ with respect to $x$ is a fraction which is not true.
2. Another method of writing the solution can be:
$\int\frac{dy}{dx} dx=\int{xdx}$
$\int{dy}=\int{xdx}$
$y=\frac{1}{2}x^2+c$
That's integrating the DE with respect to x. Is this mathematically correct? It seems like the $dx$ cancels each other in the LHS, but we are not allowed to treat this as a fraction.
My Questions are:


*

*Which of these ways of writing the solution is the most accurate?

*And which of these 2 is wrong in terms of following mathematical structure?

*When we integrate both sides of the equation, can we just put the integral sign without integrating with respect to a variable? (for example in case 1) 


It is separable equation where $ y= y(x)$. Consider the problem
$$\frac{dy}{dx}=F(x)*Q(y)$$ where $F(x)$ depends only of $x$ and $Q(y)$ only of $y$.
If $Q(y) \neq 0$ we can rewrite this as
$$\frac{y'(x)}{Q(y(x))}= F(x)$$
$$\int_{x_0}^x{\frac{y'(t)}{Q(y(t))}dt}=\int_{x_0}^x{F(t)dt}$$
Then $y(t)=s$ We will get
$$\int_{y(x_0)}^{y(x)}{\frac{ds}{Q(s)}} = \int_{x_0}^x{F(t)dx}$$
