I'll give you the big picture of the differential equations course I've taught for a few years, I'm not sure what the curriculum is at your university, but, odds are we have much in common.
First, our general goal is to solve differential equations. Depending on the problem, that may mean we seek a function whose derivatives satisfy the given DEqn, but, it may also mean we are happy to find an equation whose differential consequence is the DEqn. For example, $\frac{dy}{dx} = y$ has explicit solutions $y=ce^x$, whereas $(6x^5+2xy^2)dx+(5y^4+2x^2y)dx$ has implicit solutions in the form of the level curves $x^6+x^2y^2+y^5=c$. So, in the implicit case it's not practical to solve for $y$ as a function of $x$.
Writing a differential equation in the form $Mdx+Ndy=0$ is known as Pfaffian Form. Pfaff, a teacher of Gauss, pioneered studying differential equations as equations in differentials. There is a theorem of Pfaff which states that many DEqns of the form $Mdx+Ndy=0$ can be multiplied by $I$ such that $IMdx+INdy = dF$ where $dF = F_xdx+F_ydy$ is the total differential you likely learned in multivariate calculus. Well, this is great, because $IMdx+INdy=0$ then has $F(x,y)=c$ as a solution. Let me give you a specific instance of this,
$$ \frac{dy}{dx}+py = q $$
is the linear first order differential equation in standard form. We can write that also as:
$$ dy = (q-py)dx \ \ \Rightarrow \ \ (py-q)dx+dy =0 $$
this is not an exact equation because we cannot find $F=F(x,y)$ for which $dF= (py-q)dx+dy$. However, if we multiply by $I = I(x)$ then
$$ I(py-q)dx+Idy = 0 $$
is exact provided $\partial_x I = \partial_y (Ipy-Iq)$ or,
$$ \frac{dI}{dx} = Ip \ \ \Rightarrow \ \ \frac{dI}{I} = pdx \ \ \Rightarrow \ \ I = \exp \left( \int pdx \right). $$
Perhaps you've seen this as a device to solve $\frac{dy}{dx}+py=q$. Anyway, my point here is that the term integrating factor is used for studying exact equations and also for linear 1st-order DEs and at first glance these techinques do not seem common. Yet, the linear problem's integrating factor is actually a particular instance of the more general idea.
Ok, so, for first order problems, mostly the game is to either use calculus or algebra to wrangle the problem into something we can just integrate to see the solution. This is no longer true for higher order problems. For example,
to solve the $n$-th order homogeneous differential equation with constant coefficients we write down its characteristic polynomial, factor it, and read off the solution (this should be derived and explained, but at the end of it we have a simple prescription for solving the DE with not need for integration). For example, to solve $y''+3y'+2y=0$ write $r^2+3r+2 = (r+1)(r+2)=0$ hence $y=c_1e^{-x}+c_2e^{-2x}$ (solved, almost too easy, yes?)
to solve the $n$-th order nonhomogeneous differential equation we take the fundamental solution set of the corresponding homogeneous DEqn and run it through the variation of parameters technique.
to find that fundamental solution set of the $n$-th order homogeneous problem with possibly nonconstant coefficient functions we use power series techniques (however, in my experience, no one has patience to teach more than second order problems here)
To solve systems of differential equations we really need some insights from linear algebra, but, in a nutshell, the matrix exponential is where it's at.
Then, there are DEqns with discontinuous forcing functions, for these you want Laplace transforms.
Differential equations with singular points (say the regular type), then the Frobenius method is the way to go (this leads to many of the important special functions of mathematical physics)
I have bad news, we're still not to physics. Most of physics is PDEs (partial differential equations, this means more than one independent variable, like space and time in Maxwell's Equations or your namesake's equation). That's a whole other story. But, to understand it, you need a firm grounding in ordinary DEqns. I recommend Habermann's PDE book to get a computational grounding there. For DEqns, it's nice to have a copy of Nagle Saff and Snider's text, get the 3rd or 4th edition, they're inexpensive and have more to say than more recent editions.
In short, the labels in DEqn have to be more sophisticated than in calculus, because, we have so many different kinds of problems to solve. Moreover, as Gerald Edgar indicated, most differential equations we encounter in real world applications cannot be solved via the closed form methods I mention in this answer (or for the 100's of techniques I have not even touched here). Instead, a numerical technique is used to study the DEqns. Simulation plays a big role in what some physicists believe these days. If you can approximately build something on a computer, numerically solving relevant PDEs etc... then many will believe that's a physical possibility.
