$\def\d{\mathrm{d}}$I understand what differential equations are and what they are useful for. They're very interesting, but I'm not quite sure what is going on when actually solving one.
This is what I'm trying to solve:
$$\frac{\d y}{\d x} = 2y + 3.$$
As far as I know, this is a first-order linear ordinary differential equation. Seems fairly simple.
The first step that I am told to do is to 'separate the variables'. So I multiply both sides of the equation by $\d x$ and then divide both sides by $2y + 3$ to end up with:
$$\frac{\d y}{2y+3} = \d x.$$
So now I've got all of my $y$ terms on one side, and all of my $x$ terms on the other.
The fact that I can split apart the $\frac{\d y}{\d x}$ does seem a bit strange to me, but I'm told that it's sensible and I'm willing to believe, for now, that that's an OK thing to do.
After this, I'm told to 'integrate both sides' and I'm shown this as the next step:
$$\int \frac{\d y}{2y+3} = \int \d x$$
Normally when integrating, I would denote the 'integral of [integrand] with respect to $x$' using the standard notation:
$$\int \text{[integrand]}\,\d x$$
In definite integration, the $\int$ represents an infinite sum, and the $\d x$ represents, as usual, a very small change in $x$ that is being multiplied by the value of the function at that point in order to, as $\d x$ approaches zero, generate the exact value of the area under the curve.
For now I will just accept that we also write it in this fashion when doing indefinite integration. However, the second step in the solution to the DE confuses me because it doesn't seem to follow this notation.
If you want to integrate both sides, then that means we must be integrating $\d x$ on the right hand side. If this is true, then I would normally write it as $\int \d x\,\d x$, because it is the integral of $\d x$ with respect to $x$. However, in the solution it simply shows $\int \d x$ as if we have lost the $\d x$ that we would have put in as notation.
The other side of the equation also doesn't have a $\d y$ at the end, as I would expect it to.
My real question is: What do $\d x$ and $\d y$ represent on their own? I understand that $\dfrac{\d y}{\d x}$ represents the derivative of $y$ with respect to $x$, and that it is representing an instantaneous rate of change because it is essentially representing an infinitesimally small change in $y$, divided by an infinitesimally small change in $x$ (the gradient at a point).
However, when I see something like $\d x$ on its own, I'm not sure what the meaning of it is any more. It's an infinitesimally small change in $x$, right? How do we integrate that? What does it mean? I want to be very clear that I'm not asking someone to walk me through solving it, because the web is full of examples of solving these types of equations. I'd be much more grateful for a conceptual answer to my question.
And as an extra question: How does integrating both sides of an equation with respect to different variables make sense? Surely the two sides would no longer be equal.