When you say 'integrating both sides' isn't this sort of disguised chain rule rather than sort of an algebra technique (ie adding 5 to both sides)?
RE: Separable Differential Equations: dy/dx = ky
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Separable Differential Equations: dy/dx = ky
Can you please clarify what you mean by disguised chain rule?
The fundamental principle of algebra is applied here in that what we do to one side, we must do to the other to maintain equality.
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So you are integrating one side with respect to x and the other side with respect to y, and the justification is we 'integrate both sides' in the same way as 'we add 5 to both sides' but it isn't clear that the operations are equivalent. So something like this I can follow:
Because each time exactly the same operation occurs. Perhaps one can justify integrating with respect to different variables with the definition of the integral?
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Ah now I see. Yeah, I've skipped a step where you've shown it in equations (2) and (3). When I say that I've separated the variables, I've divided both sides by y, and multiplied both sides by dx, so that all of the y's appear on the LHS and all of the dx's appear on the RHS after cancelling.
Note that dy/dx can be regarded as the quotient of 2 differentials, which means they are separable - they don't have to stay together.
I think I've explained it better in the video, as well as my previous video, rather than the text I've posted.
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