-4x+y+(x+4y)y'=0
give value for k in which differential eqn is exact
10x^19*y^2+kx^20*y+(x+y)*x^20 dy/dx =0
dy/dx=2x^2-3x-2+(2/x)*y
dy/dx=(6-(x+3)y)/x+2
$\displaystyle \frac{dy}{dx}=2x^2-3x-2+\frac{2}{x}y $
$\displaystyle \frac{dy}{dx}-\frac{2}{x}y=2x^2-3x+2$
so we need our integration factor
$\displaystyle I(x)=e^{\int \frac{2}{x}dx}=e^{2 \ln(x)}=e^{\ln(x)^2}=x^2$
$\displaystyle x^2\frac{dy}{dx}-2xy=2x^4-3x^3-2x \iff \frac{d}{dx}\left[ x^2y \right]= 2x^4-3x^3-2x $
integratin we get
$\displaystyle x^2y=\frac{2}{5}x^5-\frac{3}{4}x^4-x^2+c$
$\displaystyle y=\frac{2}{5}x^3-\frac{3}{4}x^2-1+cx^{-2}$