Hi BlackPenRedPen! I would like to ask you a simple question:
Every time I watch a video when you're solving a differential equation, where the general solution involves ln | "a function in terms of y" | = "a function in terms of x" + C1 I have the same question. We're trying to isolate y = f(x), let me put all the steps in a picture:
And then from here we continue isolating y... My question is: When you carry out the absolute value, why are you merging the +- C2 into another constant with positive sign? I mean, of course that -C2 = C3, or e^(C4) = C5... But here we have two possible values! +C2 and -C2, both at the same time! When you merge it in C3 we lose one of the mathematical solutions... OK, according to initial conditions or the context of the problem where the diff equation is involved, maybe one of these solutions is impossible. For instance, when solving quadratic equations, we could have positive and negative answers but if we're talking about lengths of a triangle, we cannot have negative lengths!
I think we should keep the +- when solving differential equations, and then:
- Leave it as how it is if we just solved the differential equation for fun (so it is not involved into any context). Or
- When solved, plug in the initial conditions to solve for the constants (both positive and negative cases) and see whether one of the obtained solutions has no sense... Then, at this point, remove the wrong solution.
Please, I think you can make a video on this! It's sounds quite interesting...
Appreciate you time, best regards!