Imagine that you are taking a stroll home after a long day spent doing Real Analysis and pitting proving skills against tea buddies. It is very late into the night and your wife is getting impatient, so you decide to take a shortcut home through the deep, meandering alleyways through the city's underbelly.
It is your first time here; you do not know yet that by entering the slums, you have thrown Real Analysis to the four winds and yourself into that desperate quagmire which is Differential Equations.
Before you know it, you have angered the denizens of the alley. You meet your first adversary.
This is a passing drunkard who just needed someone to blab at. Fortunately, you know that these drunkards can easily be subdued using Leibniz's notation of the derivative, dy/dx.
The poor fellow has scarcely put up a fight. He does not even try to get up again, so you roll him to his side, lest he choke on his own vomit.
As you advance, several more drunkards appear (have you been walking close to a club?). They do not look as soft as the first one, but you pwn them anyway by using simple substitutions and then invoking the power of Leibniz.
As you retreat further away from the hypothetical nightclub, your assailants become less and less benign. News seems to have leaked that a novice mathematics student has wandered into D.E. territory, and the first bandits move in to the plunder: the First Order Linear gang with the Bernoulli and Ricatti clans.
However, they have sorely underestimated your prowess; obsessed as you are with logical clarity and intuition, you are not above borrowing the ideas of the masters without questioning. The raiding party learns this the hard way.
The defeat of the Linear First Order sends shock-waves through the bandit community. Clearly, this guy knows a thing or two, even if all he knows is engineering math! Not admitting defeat, the bandits send out a second detachment.
Unfazed, you tackle the leader of the pack first: the homogeneous equation. You subdue him with a risky but deadly substitution
and reduced the pack leader to a shapeless heap of general solutions.
The rest of the pack are scared shiteless, because by the Superposition principle, they are already halfway into utter defeat. Working down the hierarchy with the Method of Undetermined Coefficients, you make sure that the gangsters with the polynomial, exponential and trigonometric forcing functions wish they had never been born. In that very order.
This is an ancient and arcane martial art, the underlying wisdom to which you have very little idea of. But never mind, right now it is enough to know all the moves. Besides, you have to confront your next opponent, the real boss of this line-up, the all-encompassing Nonhomogeneous Second Order Monstrosity with Nonconstant Coefficients.
You are forced now to use a tactic which is so tedious, your trainers expect you to refer to the manual every time the need arises.
Heaving a sigh of regret, you recall that you have accidentally left your prize weapon at home, or was it at the tea party? Where is your Laplace Transform when you most needed it?
In any case, you eventually bring down the boss with the Variation of Parameters, a method even more inscrutable than the last. In fact, it is so arcane that it is not even clear why it is called "Variation of Parameters".
The coast is clear, at least for now. You look wistfully at the Farlow et al. (2007) 2nd ed. which has guided you through this tough journey. Unfortunately, you are still very far away from home. Feeling absolutely worn out after many fights, you drift to sleep to wake up in the real world that you call your own.
