Well, a somewhat facetious answer would be
Partial Differential Equations
--> Can we use Viscosity Solutions and/or Maximum Principles?
Yes? Do it and publish. End.
--> Is the equation/system integrable?
Yes? Write down the Lax Pair/conservation laws and publish. End.
--> Integrate by parts and publish. End.
(NB "integrate by parts" include weak solutions, finite element methods, energy methods...)
A more serious answer is that there are almost as many "methods" to solve equations as there are "types of equations". For ODEs the Wikipedia entry does a reasonable job illustrating this: the taxonomy of equations are generally by the method with which it can be studied. Unfortunately, the classification by method generally bear little resemblance to the physical attributes of the equations (linear, non-linear, what kind of nonlinearity, the degree, the principle part of the differential operator, etc).
A fantastic classical illustration for this is the Korteweg--de Vries equation describing shallow water waves. The equation itself looks rather deceptively simple:
$$ \partial_t u + \partial^3_x u + u\partial_x u = 0 \qquad \qquad (KdV)$$
for some $u$ defined as a function of $t$ and $x$. As Dick Palais explained in his Bullentin article on solitons, one might be tempted to guess at the large-time behaviour of solutions of this equation, based on its simpler cousins:
$$ \partial_tu + \partial^3_x u = 0 \qquad\qquad (3disp) $$
$$ \partial_t u + u \partial_x u = 0 \qquad \qquad (Bgr) $$
The first equation is obtained by retaining the "leading order term" in the $x$-derivatives (which would be what would happen if you attempt to classify equations by order) and dropping the non-linear "lower order term". The second equation is obtained by an "ODE in time" procedure, which suggests, roughly speaking, that resonance is the principal thing to worry about for evolution equations (and resonance manifests itself in non-linear interactions of a function with it self, the higher the algebraic power [more times you multiply a function by itself], the worse the resonance feedback would be).
Equation (3disp) is called a dispersive equation. The main idea is that because of the cubic derivative in space, if we perform Fourier analysis for this equation, we see that the different frequency components travel and different speeds. Physically this means that if you start with a wave packet as initial data, the packet will spread in physical extent will decaying in amplitude. This essentially guarantees that all reasonable initial data (which can be allowed to be large) will eventually settle down to a constant.
Equation (Bgr) is the Burgers' equation, a sort of prototypical nonlinear wave equation. Being a first order equation, one can solve this using the method of characteristics, and qualitatively we see that the equation describes a traveling wave, whose traveling speed is proportional to its amplitude. Now this gives a problem. If you have the initial data such that a high amplitude portion of the wave is behind a low amplitude portion of the wave, eventually the high amplitude portion will run up right against the low amplitude portion (since it is moving faster and catches up). When this happens, at that point in space, we have a discontinuity of the wave (a sudden drop of amplitude from really high amplitude to really low amplitude). This signals the formation of shock waves. One can also see this explicitly by differentiating the equation in space: let $v = \partial_x u$, we see that $v$ solves the equation
$$ \partial_t v + u \partial_x v + v^2 = 0$$
So using the method of characteristics, we see that allong the integral curves of the vector field $\partial_t + u\partial_x$, $v$ solves the equation $\partial_s v = - v^2$ which can be integrated to show that if $v$ is initially negative at any point in space (if the slope of $u$ is negative, which means higher amplitude behind lower amplitude), the solution must blow up in finite time.
So if you just follow the "genealogy" and combine the ideas from the two simplifications into the final (KdV) equation, you may be tempted to say that perhaps there will be regimes in which the behaviour of each of the simplifications dominates. As it turns out, the tug-of-war between blow-up of (Bgr) and dispersion of (3disp) gives rise to a completely different long time dynamics! That is: if you start with arbitrary reasonable initial data, the solution eventually breaks up into a finite number of coherent wave-packets traveling along the real line. In particular, each of those coherent wave-packets do not disperse (by itself), but as an ensemble some wave-packets would move faster than others and so as a whole the solution spreads out.
This is not to say, of course, there are no stable features. The physical feature of being a second-order elliptic equation is fairly stable to lower order perturbations, and to various non-linearities. And lo-and-behold, volumes and volumes have been written about this type of equations. And presumably there are handbooks, such as the one Hans Lundmark mentioned in his comment, that describes large portions of the "known" differential equations and methods used to solve them. I only hope the above serves as a cautionary tale against treating as-of-yet unstudied equations purely by analogy with things we already know.