Question

Show that any separable equation
M(x) + N(y) yَ = 0
is also exact

Answer 1

Not any separable equation is exact.. well let's see:
M(x) + N(y) (dy/dx) = 0
Multiply across by dx
M(x)dx+N(y)dy=0.

Answer 2

that equation is going to be exact if and only if:
Mx = Ny

Answer 3

then how im gonna proof that ??? its a problem in Willian E. Boyce's Book
Elementary Differential Equations \ 7th Edition

Answer 4

No just take the function F = int M(x)dx + int N(y)dy, so that pdF/dx=M(x) and pdF/pdy=N(y). So so by definition the equation is exact.

Answer 5

i still dont get it.. can u type it as Equation formula please :)

Answer 6

I just mean for a diff eqn to be exact it needs to be of the form Mdx+Ndy=0 with a function F(x,y) s.t. \[\delta F/\delta x = M\] and \[\delta F/\delta y = N\] (I can't find partial d).

So if this case we already have M(x)dx+N(y)dy=0 so all we need is a function F. The function I wrote above works as this function, since the int Ndy disappears when partial diff by x and vice versa. So F works and the eqn is exact.

Answer 7

Thanx alot .. helpful :)