I am a 1st yr. College student. My question is :: Generally, while dealing with calculus problems ( differential equations, limits, integration ,etc.. ), we work with a large no. of transcendental and algebraic functions and expressions, each of them having specific domains and ranges. Thus in the course of the problem, we are supposed to check the validity of each and every step and ensure that the information regarding the domains is preserved. Specifically, we should check that each and every division performed in the solution should be valid( with non zero denominator). However, while reading and searching through STANDARD TEXTS on calculus, I find that this practice is not EXPLICITLY followed, i.e. in the solution, blatantly expressions are divided to proceed through ( without justifying the division ). For e.g. A step in the solution of a d.e. requires the division of a particular expression by ($\frac{\mathrm dy}{\mathrm dx} - 6y^3+ 7$) to simplify and proceed on the solution path. What I observe( in texts, or in people solving) in such cases is that the division is carried forth without keeping the provision for the case when $\frac{\mathrm dy}{\mathrm dx} = 6y^3-7$. While doing problems I always try to check such cases which greatly reduces my speed in addition to the overburdening headache of keeping the list of exceptions.
Hence, even while solving simple differential equations or manipulating trigo expressions, I tend to take much more time, while others get exactly the same answers without the additional checks.
Should I continue with the checks or move on without them or only a few of them( e.g. checking the final answer with the initial conditions, constraints, etc. ). I would be thankful for any clarification or suggestions to my problems. Thanks for bearing with me in this obscure, long thing.
Yes, please keep on worrying about checking all those special cases!
The regrettable fact that some textbook authors are sloppy (when solving separable ODEs, in particular) doesn't make it right.