Given $r>0$ and $f:\mathbb{R}^n\to \mathbb{R}$, $d_rf$ is the function defined by \begin{equation}d_rf(x_1,x_2,\dots,x_n)=f(rx_1,rx_2,\dots,rx_n)\end{equation} and is called the $r$-dilation of $f$, and $f$ is said to be homogeneous of degree $D$ if $d_rf=r^Df$. Similar notion is defined for distributions by duality.
I do not know much about differential equations, and do not actually see why this notion is so important. Wikipedia says homogeneous functions are good because equations involving them can be solved by separation of variables. But I am not sure how that works and how that is related to the homogeneous degree of a function.
Can someone explain this? Thanks!