DE notation

[Vol]

is the definition of the differential equation in some books. My question is, why is it y(x) and not just y? Can't understand exactly why.

 

[MarkFL]

I would say it just explicitly shows that \(y\) is to be treated as a function of \(x\).

 

[Vol]

OK. Here is another question then. With exact differential equations, sometimes the equations is put like this:
M(x,y)dx + N(x,y)dy = 0 but other times it is put like this:
M(x,y) + N(x,y)y' = 0.
How come in the second equation you have y' in the second term? Why the difference?

 

[HallsofIvy]

I am puzzled by this. You are asking about the differential equation \(\displaystyle M(x,y)dx+ N(x,y)dy= 0\) but appear to be saying that you do not know what "dx" and "dy" mean!
The "dx" and "dy" here are differentials and satisfying \(\displaystyle dy= f'(x)dx\). That is the same as \(\displaystyle \frac{dy}{dx}= f'(x)\). Given the equation \(\displaystyle M(x,y)dx+ N(x,y)dy= 0\), divide through by dx to get \(\displaystyle M(x,y)+ N(x,y)\frac{dy}{dx}= M(x,y)+ N(x,y)y'= 0\).

 

[Vol]

Oh, I get it. But why divide through by dx? How does that help?

 

[HallsofIvy]

?? In order to get y' which is what you asked about!