Separable Equation - Notation Question w/ Infinitesimals

[neporuoydnim]

Please see below link for the two different styles of solving a separable equation.
http://en.wikipedia.org/wiki/Separation_of_variables#Ordinary_differential_equations_.28ODE.29

Which one is more proper? Why? My DE teacher told me that strictly speaking it's wrong to use the first method because infinitesimals do not have the same properties as other algebraic quantities and therefore can't be multiplied across then canceled so simply.

Thank you for your time. This is driving me crazy. I don't like to do my homework until I figure out the details.

 

[AlephZero]

I thnk you need to get familiar with the first notation, whatever your teacher says about it, because that's what nearly everybody uses nearly all of the time.

Notation is just making marks on a piece of paper (or a computer screen). What really matters is that you are doing valid mathematical operations - though of course a good notation makes it easy to write down valid operations, and avoid doing something wrong by accident.

You can think of the second version on the Wiki page as a way to prove that the notation in the first version really does correspond to correct math. You don't need to prove a math theorem over and over again every time you use it. Once it's proved, it stays proved for ever. You don't need to write out all the details of the second method for every ODE that you solve, any more than you need to prove Pythagoras's theorem again every time you use a right angled triangle in your math work.