Question: show it is homogeneous and solve it dy/dx = (x+3y)/(x-y)
Solution： dy/dx = (1+3y/x)/(1-y/x) in this form, which shows it is homogeneous
let u = y/x, y = ux
differentiate on both sides with x, we get
dy/dx = u + xdu/dx
from above, dy/dx = (1+3u)/(1-u)
then (1+3u)/(1-u) = u + xdu/dx
xdu/dx = (1+3u)/(1-u) - u = (u+1)^2/(1-u)
(1/x) dx = ((1-u) / (u+1)^2) du
integrating on both sides, we get
ln|x| = - 2/u+1 - ln|u+1| + C
-ln|x| - 2/(y/x+1) - ln(y/x+1) + C =0
- ln(y+x) - 2x/y+x + C = 0
