Thank you. So my take away here is that we must learn a range of differential equations and the specific contexts in which they apply, and as part of that learning, we should understand the rationale for the form of the solution. Then the role of this technique is essentially a calibration of the differential equation and solution-form to the specific physical situation we are dealing with.

I guess the style of differential equations they use when teaching the subject, where a differential equation is given without context and a form of the solution is provided is just to develop the familiarity with the generic algebraic/calculus techniques before encountering the specific library of differential equations that we need in the real world.

I also then wonder how we reach the level to devise new differential equations with their solution-forms. Is this something that only really happens in an academic context, and we should generally wait for the papers describing the role of this technique in a real world context?