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)