What does P(x,y)dx + Q(x,y)dy = 0 mean?

Question

More specially, can you explain what P(x,y)dx mean?

How do you write it in differential form?

Answer 1

this is an exact differential equation provided that the derivative of P(x,y) wrt x is the same as the derivative of Q(x,y) wrt y.

Here's a great page that explains it much better than I ever could:

Answer 2

Here, P(x,y) dx represents the component of the function that is concentrated in the x-coordinate. That is, for every dx that you traverse along the x-direction, the value of the function increases by P(x,y). This is differential form, and it allows you to take a multivariate function and break it up into two orthogonal components (dx and dy), for simplicity's sake.

Answer 3

Have you missed out integral signs?
The reason I ask is that dx means with respect to x and dy means with respect to y.
P(x,y) and Q(x,y) are functions of x and y.
eg 3x²y and 5xy³ say, which then becomes:-
∫ 3x² y dx + ∫ 5xy³ dy = 0

Answer 4

Just as "1" is the multiplicative identity (ie a X 1 = a) zero is the additive identity (a + 0 = a). This concept makes a lot of other mathematical theorems work.

Answer 5

The sum of the derivative functions P and X is zero.

Their "slopes" are the same except for opposite sign, so when added equal zero.

It is written in differential form.

Answer 6

does P(x,y)dx mean product rule
& Q(x,y)dy mean quotient rule ??????????