I have quite some trouble thinking why we are allowed to manipulate differentials as we see fit when solving differential equations. I usually think of the derivative as the fundamental object upon which differentials are based. With this in mind I wince when I see derivatives appear separately, as in separation of variables when solving differential equations.
For example, consider [itex] \frac {dy}{dx} = ky [/itex]
We would "separate" the variables as follows: [itex] \frac {dy}{y} = k dx [/itex]
And then we would integrate both sides... [itex] \int \frac {dy}{y} = \int k dx[/itex]
What I don't understand is what allows us to separate the variables... since when are we allowed to multiply and divide both sides of an equation by a differential? I thought differentials were not like normal numbers, and you're not allowed to play with them unless some theorem specifically allows you to do so?
Also, why are we allowed to integrate both sides of the equation with respect to different variables?
I appreciate any help/advice. Thanks!
BiP
