Hi, I'm new here and I have a question about exact differential equations and integrating factors.
I was looking at the following online pdf about special integrating factors:
http://www2.fiu.edu/~aladrog/SpecialIntegratingFactors.pdf
and I need help understanding the first example problem, which I'll print here.
1) (4xy + 3y2 – x) dx + x(x + 2y) dy = 0
So I get it up to the point where you multiply by the integrating factor you find, x2, to get:
(4x3y + 3x2y2 – x3) dx + (x4 + 2x3y) dy = 0
But I don't understand why the example says this is exact, since when I find the partial derivative of M and N, they don't match.
I get:
Mx=12x2y + 6xy2 - 3x2
Ny=2x3
Which don't equal, so I'm not really sure what to do from here.
And why did the example choose to group it like this? How does grouping it in specific ways help?
(4x3y dx + x4 dy) + (3x2y2 dx + 2x3ydy) – x3 dx = 0
Thanks for any help in advance!
I was looking at the following online pdf about special integrating factors:
http://www2.fiu.edu/~aladrog/SpecialIntegratingFactors.pdf
and I need help understanding the first example problem, which I'll print here.
1) (4xy + 3y2 – x) dx + x(x + 2y) dy = 0
So I get it up to the point where you multiply by the integrating factor you find, x2, to get:
(4x3y + 3x2y2 – x3) dx + (x4 + 2x3y) dy = 0
But I don't understand why the example says this is exact, since when I find the partial derivative of M and N, they don't match.
I get:
Mx=12x2y + 6xy2 - 3x2
Ny=2x3
Which don't equal, so I'm not really sure what to do from here.
And why did the example choose to group it like this? How does grouping it in specific ways help?
(4x3y dx + x4 dy) + (3x2y2 dx + 2x3ydy) – x3 dx = 0
Thanks for any help in advance!