Homogeneous definition for first order differential equation and higher order differential equations?

and for higher order, it is https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_linear_differential_equations

So, using the definition for higher order can we prove a first order differential equation is homogeneous?

For an example; $$4\frac{dy}{dx}+y=0$$ is homogeneous considering definition for higher order linear D.E. But using the original definition for homogeneity of first order we get $$\frac{dy}{dx}=\frac{-y}{4}$$ which does not seem to be homogeneous.

I saw definition for higher order being used for first order as well, so got confused. Here is a snap of the video, where N=non homogeneous and H=homogeneous.

1 Answer

An equation is homogeneous if it is linear in $$x$$ and its derivatives. Linear here means not affine, but in the sense that $$x=0$$ is an element of the solution space.

It does not matter what equivalent form of the equation you use (as long as it stays linear).