differential equations: integrating factor?

Question

the question says to find the integrating factor.

(12+5xy)dx + (6xy^(-1) +3x^2)dy =0

the little ^ indicate powers.... wasn't sure of the proper notation.

I can do the normal ones but this one seems to never cancel anything out......

any help apreciated

Answer 1

First let's re-arrange this equation.

(12+5xy) dx = -(6xy^-1+3x^2) dy =>

(12 + 5xy)/(6x/y+3x^2) = -dy/dx
(12+5xy)*y = -(6x+3x^2) dy/dx
0= 12 y + 5x y^2 - (6x+3x^2) dy/dx Non-linear

I don't see a way to get your equation into the form f(x)=g(x)y+y' in this equation the integral of e^g(x) would be your integrating factor.

Good Luck.

EDIT

The Exact Equation
M(x,y) dx + N(x,y) dy =0 leads to Integral M dx + integral (N- d/dy integral M dx) dy so...

integral (12+5xy) dx + integral (6x/y +3x^2 - d/dy integral (12 + 5xy dx) dy

12x+5/2x^2y+ integral (6x/y +3x^2 - d/dy (12x + 5/2x^2y) dy)
12 x + 5/2 x^2 y + integral (6x/y +3x^2 - 5/2x^2) dy
12x + 5/2 x^2 y + 6x ln y + 1/2 x^2 y = C

12 x + 3 x^2 y + 6x ln y = C I'll let you solve for y