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For first order https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_first-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. enter image description here

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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).

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