Divide through by 8x:
dy/dx+9y/(8x)=x2ln|x|y3/8. This conforms to a Bernoulli DE.
Let v=y-2, then v=1/y2, y2=1/v, y=v-½, dy/dx=-½v-3/2dv/dx.
The DE becomes:
-½v-3/2dv/dx+9v-½/(8x)=x2ln|x|v-3/2/8.
Multiply through by v3/2:
-½dv/dx+9v/(8x)=x2ln|x|/8, dv/dx-9v/(4x)=-¼x2ln|x|.
Multiply through by x-9/4:
x-9/4dv/dx-9vx-13/4/4=-¼x-¼ln|x|,
d(x-9/4v)/dx=-¼x-¼ln|x|,
x-9/4v=-¼∫x-¼ln|x|dx.
Let u=ln|x|, du=dx/x.
Let dv=x-¼dx, v=4x¾/3,
∫x-¼ln|x|dx=4x¾ln|x|/3-(4/3)∫x-¼dx=4x¾ln|x|/3-x¾
x-9/4v=-x¾ln|x|/3+x¾/4=¼x¾(1-4ln|x|/3)+C, where C is a constant.
v=¼x3(1-(4/3)ln|x|)+Cx9/4, y2=4/(x3(1-(4/3)ln|x|)+Kx9/4) where K=4C. -(4/3)ln|x| can be written ln|1/x4/3| or ln|x-4/3|.
Solution to be confirmed...