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If I remeber right you can do that, because there’s a longer and proper way that arrives at the same conclusion
Yes, and what is aimed for is separation of variables to find a solution of this differential equation. (The differential operator d should be printed as upright letter btw)
There’s no reason it shouldn’t work.
dy/dx is the same as (y1 - y2) / (x1 - x2) as the distance between the two points approaches zero. “dx” and “dy” aren’t very useful measurements on their own though.
Worse, I like to cancel out the "d"s.
Yes, yes we can!
You can’t multiply both sides by dx in much the same way you can’t differentiate a duck. That said, even pure mathematicians sort of think of it that way as a useful shorthand.
Can’t you just use infinitesimals and then actually multiply them? It never results in an invalid operation with the normal dx, only the one with the fancy d (forgive my lack of terminology knowledge)
In (d/dx)f(x), d/dx is a symbol that means the derivative of f with respect to x. It’s not a division of two variables. But, the reason the symbol is useful is that you sort of can multiply the dx in some situations.
I understand that it’s a symbol, not a fraction, and that the top and bottom are linked and not separable. But, you can also use an equivalent infintesimal fraction dy/dx with the actual infintesimal values dy and dx being manipulatable. If I’m wrong, you’ll be able to find an example that doesn’t work (without using partial derivatives-- those actually can’t be cancelled).
