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
Exact differential equation,
yn[(2y2)dx−x((2x3)+y)dy]=0Where,M(x,y)=2y2+nN(x,y)=−2x4yn−xy1+nThen,My=Nx2(2+n)y1+n=−8x3yn−y1+n(4+2n)y1+n=−8x3yn−y1+nThere doesn’t exist any n for whichynis a integrating factor.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.}yn[(2y2)dx−x((2x3)+y)dy]=0Where,M(x,y)=2y2+nN(x,y)=−2x4yn−xy1+nThen,My=Nx2(2+n)y1+n=−8x3yn−y1+n(4+2n)y1+n=−8x3yn−y1+nThere doesn’t exist any n for whichynis a integrating factor.
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