Having trouble understanding the concept behind seperable equations

[mangopear]

So I'm in my first differential equations college class, and we just learned about separable equations. Now, I have no trouble solving these equations, but I'm having trouble understanding what is actually happening when we solve a separable equation.
For example, the generic separable differential equation is N(y)dy/dx=M(x).
Then you simply cancel out the dx and pop it on the right side resulting in two clean integrals. N(y)dy=M(x)dx which you can then solve easily.
Here's my problem. I don't understand how we can just split a derivative and algebraically separate dx from dy. I understand that we're not actually canceling out the dx, but I'm not sure what exactly is happening. Could somebody please elaborate?
Thank you!

 

[Ishuda]

So I'm in my first differential equations college class, and we just learned about separable equations. Now, I have no trouble solving these equations, but I'm having trouble understanding what is actually happening when we solve a separable equation.
For example, the generic separable differential equation is N(y)dy/dx=M(x).
Then you simply cancel out the dx and pop it on the right side resulting in two clean integrals. N(y)dy=M(x)dx which you can then solve easily.
Here's my problem. I don't understand how we can just split a derivative and algebraically separate dx from dy. I understand that we're not actually canceling out the dx, but I'm not sure what exactly is happening. Could somebody please elaborate?
Thank you!

Let
F(y) = \(\displaystyle \int^y N(y) dy = \int^x M(x) dx\)
Take the derivative wrt x across and use the chain rule.

 

[HallsofIvy]

The derivative, strictly speaking, is NOT a fraction but it is the limit of a fraction, the difference quotient. As a result, we can prove many "fraction-like" properties by going back before the limit, using the fraction property, then taking the limit again. For example, the "'chain rule", \(\displaystyle \frac{df}{dx}= \frac{df}{dy}\frac{dy}{dx}\) where f is a function of y and y is a function of x. We cannot prove that by just "cancelling the 'dy's" but we can prove it by, as I just said, going back before the limit.

In order to be able to use the fact that a derivative can be "treated like a fraction", we define the differentials with "dx" being purely symbolic and dy defined as f'(x)dx. That is typically done in a first Calculus class. Once we have done that, "separating" derivatives in order to solve separable equations is valid.

 

[mangopear]

OK, that makes more sense. So we're not actually treating dx/dy like a fraction, but we're using proofs to rewrite it as fractions for ease of use?
Thanks for the replies by the way!

 

[HallsofIvy]

Make sure you understand my other point- that we can then define "dy'' and "dx" separately so that "dy/dx" is a fraction.