Taking that second question,
$r' = i - nr - jq$
and differentiating gives
$r'' = -nr' - jq' = -nr' - j(k-\frac{m}{r})$
or in other words
$r'' + ar' + \frac{b}{r} = c$
which is a much simpler differential equation only one variable. I think that you could probably solve this with power series or clever guessing, but it needs to be worked out.
Your question would deserve a very long and complicated answer. To summarize, you may think of the development of the concept of function. At the very beginning, a function was an equation, i.e. a formula $y=\ldots$ written by elementary "atoms" (powers, logarithms, sines, cosines, etc). It then become clear that an "abstract" idea of function was more useful than that: sometimes there is no finite combination of "atoms" to write down a function, and yet you can study its properties, like $\int e^{x^2}\, dx$.
In the realm of differential equations, most equations do not have explicit, elementary, solutions. Solutions are found by integrating complicated expressions, sometimes they are found implicitly, and there is no hope to write a solution as $y=\ldots$
I am not an expert of history, but the breakthrough happened when functional analysis began to develop. A differential equation is simply (!) and equation whose unknown lies in a function space. From this viewpoint, it is more important to study qualitative properties of the solutions, rathen than to write down a crazy formula with power series, special functions, and so on.
I believe that the study of differential equations was a stimulating problem for a large part of modern mathematical analysis, in the last 150 years (more ore less). Just think of nonlinear functional analysis, that was born essentially to solve differential equations as fixed-point or variational problems.
I cannot say if there was a precise reason, in the hstory of mathematics, why mathematicians had to abandon explicit solutions. The necessity probably grew up slowly.
Best Answer
In general, these equations do not have closed-form solutions, or even an integral formula for solutions. In the homogeneous case ($d=0$) where $a$, $b$, $c$ are rational functions there is an algorithm due to Kovacic to decide whether there are "Liouvillian" solutions (roughly speaking, those are solutions that can be expressed in terms of the exponential function, algebraic functions and integrals). For example, $y'' + (x^3+1) y = 0$ has no Liouvillian solutions (and, as far as I know, no closed-form solutions of any kind). However, this is not something for beginners to deal with.