Answer to Question #209719 in Differential Equations for caleb

Question

The integrating factor of the given DE are already below. What is the value of "n".

(2y^2)dx-x((2x^3)+y)dy=0

IF=y^n

Answer 1

Exact differential equation,

$y^n[(2y^2)dx-x((2x^3)+y)dy]=0\\ Where,\\ M(x,y)=2y^{2+n}\\ N(x,y)=-2x^4y^n-xy^{1+n}\\ Then,\\ M_y=N_x\\ 2(2+n)y^{1+n}=-8x^3y^n-y^{1+n}\\ (4+2n)y^{1+n}=-8x^3y^n-y^{1+n}\\ \text{There doesn't exist any n for which}y^n\text{is a integrating factor.}$