The basic definition of homogeneous is that it makes no difference if you multiply every variable by some (nonzero) number $k$. If $f(x,y)$ is a function of two variables, then we'd call it "homogeneous" if $f(kx,ky) = k^df(x,y).$ In differential equations, the $d$ is usually 0. The function $f(x,y) = x^2+y^2$ is homogeneous of degree 2 because $f(kx,ky) = k^2f(x,y).$
This definition manifests itself in two ways in DE. In 1st order equations, a DE is "homogeneous" if we can write $y^{\prime} = f(x,y) = g(y/x)$. See that in this case, $f(kx,xy) = g(ky/kx) = g(y/x) = f(x,y)$.
In 2nd order equations, we call a linear DE "homogeneous" if there is no lone function of $t$. Such as $y^{\prime\prime}+4y^{\prime} + 3y = 0$. See that if you replace $y$ by $ky$ in this equation, the $k$ factors out and you can divide it away, leaving you with the equation you started with.