斉次な微分方程式

最近微分方程式の解き方を学んでいます。一つな問題は斉次な微分方程式です。It seems strange that, although an equation is homogeneous, it turns out that the variables cannot be separated to allow simple integration to occur. 斉次と言うのは方程式をM(x,y)ｄｘ＋N(x,y)ｄｙ　＝０　の形にする。
ｘとｙの代わりにｃｘとｃｙを入れ替えればM(cx,cy)＝ｃ＾ｎM(x,y)とN（ｃｘ、ｃｙ）＝ｃ＾ｎN（ｘ、ｙ）。
However, a homogeneous diff eq which cannot be solved satisfactorily (it seems) is
x(dy/dx)-2y = -x
Solving this equation by assuming it is homogeneous (using y=vx substitution) yielded a different result than solving it by assuming a nonhomogeneous equation (substituting y=uv and initially setting the v coefficient to 0 in u(dv/dx)+v(du/dx+Pu)=Q.
斉次じゃない常微分方程式を解けば積分因数を使用する。
以上な方程式は斉次にしても不斉次にしても正解の算出が出来るらしい。Solving x(dy/dx)-2y=-x using the y=vx change of variable, integrating the separated variables (using (dx/x)=(dv/(f(v)-v))) and taking the exponent of both sides (as the integral outputs a logarithm), the result is:
y = c(x*x)+x
This agrees with the WikiHow example, which does not assume homogeneity.