How to determine if the differential equation is homogeneous?

Question

I'm confused about if the equation $y''+\sin(x+y)=\sin x$ homogeneous? By what our teacher said, if we plug $y=0=$ const into the equation and it matches on both sides, then it is homogeneous. By this criterion, it is homogeneous, am I right?

Answer 1

The ODE already belongs to inhomogeneous because of the $\sin x$ term.

Even letting $u=x+y$ ,

Then $u'=1+y'$

$u''=y''$

$\therefore u''+\sin u=\sin x$

It still belongs to inhomogeneous.