How would you not find non-homogeneous differential equations in real life? Springs! Hooke's law dictates that springs will usually vibrate by non-homogeneous second order differential equations.
Was that supposed to be grammatical?
The way you don't find non-homogenous differential equations is by not looking for them; for example, if your area of interest is not springs, but population dynamics, then Hooke's law is irrelevant.
Which means a typical autodidact will not explore solution techniques for such equations....
They pass if they get the job done well.
Which means, in this case, that they solve the population dynamics problems well, and they lack the application breadth of a typical Diff Eq veteran.
Ergo, they "pass" only in their own mind; not in any sense that would pass academic rigor.
Which is the fundamental problem in the first place.
You can assume someone knows what they're doing if they're doing their job well.
And if they do their job badly, will they reimburse their employer for the wasted time and trouble?
Most employers prefer to evaluate whether or not someone can knows what they're doing before offering them a job to do well -- or not.
If they can do their job well and they don't have all the coursework then that coursework wasn't important.
When I learned English, I never formally learned a future tense! However, I learned how to express ideas grammatically. I just never thought of it as a "future tense". Roman history is roughly a thousand years long and then some(approximately 500 BCE to 500 CE)!!! Caligula wasn't a major historical figure in many relevant aspects of Roman history. In fact, several Roman historians only gave cursory information and opinions of his reign, among them Pliny the Elder.
Congratulations. You've just proven my point in spades.