HOMOGENEOUS DE: (xylny+ylnx)dx+x(lnylnx)dy=0

1. (x - ylny + ylnx) dx + x(lny - lnx) dy= 0
2. (x csc y/x - y) dx + xdy=0
3. (x^2 + 2xy - 4y^2) dx - ( x^2 - 8xy - 4 y^2)=0
4. x^y ' = 4x^2 + 7xy + 2 y^2

Solution to No. 1
(xylny+ylnx)dx+x(lnylnx)dy=0

[xy(lnylnx)]dx+x(lnylnx)dy=0

[xyln(yx)]dx+xln(yx)dy=0

Let
y = vx
dy = v dx + x dv

[xvxln(vxx)]dx+[xln(vxx)](vdx+xdv)=0

(xvxlnv)dx+(xlnv)(vdx+xdv)=0

xdxvxlnvdx+vxlnvdx+x2lnvdv=0

xdx+x2lnvdv=0

dxx+lnvdv=0

lnx+(vlnvv)=c

lnx+yxln(yx)yx=c