Question

obtain a family of solutions for \[(x-2y)dx + (2x+y)dy = 0\]

Answer 1

put it in this form first:
dy/dx=-x+2y/2x+y
then devide by x the right side
dy/dx=-1+2(y/x)/2+(y/x)
make variable change
y/x=u
y=ux
dy/dx=u+x(du/dx)
substitute in the DE
and solve by separating variables

Answer 2

why cant you sub from the start?

Answer 3

you could, just is more clear other way

Answer 4

i see

Answer 5

since you don't mess with differentials

Answer 6

is u + x(du/dx) = udx + xdu?

Answer 7

yes

Answer 8

great..thanks