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A function of two variables is said to be homogeneous of degree n if there is a constant n such that $$f(tx,ty)=t^nf(x,y)$$ for all $t, x, y$ for which both sides are defined.

A differential equation of the form

$$M(x,y)dx+N(x,y)dy=0$$

is homogeneous if $M$ and $ N$ are homogeneous of the same degree. You can solve homogeneous equations by turning them into separable equations using the substitution $$y=vx.$$

My question is, why is this substitution mathematically valid? I understand that for every $y$ and $x$ there exists a $v$ such that $y=vx$, but using this substitution seems to suggest that there exists a $v$ such that $y=vx$ for every $x$, or that you're just avoiding dealing with the differentials in $x$ by inventing a variable $v$ and varying that instead of $x$ or something.

I understand how to proceed algorithmically once the substitution is made, and how, if you assume it all makes sense, that it can lead to nice tidy solutions, but I don't really understand--analytically, I guess?--what we're actually doing by making this substitution.

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    Think of $v$ as being the function $y/x$. It maps $x$ to the value $y/x$. That way, may be you are able to understand the substitution. $v$ is a function of $x$. – 2017-01-07

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Analytically: Well, first of all finding general solution of a Differential equation in homogeneous form can be really tedious by directly integrating, because sometimes we will not get explicit solution (i.e in the form y=y(x)). (Note:To get explicit solution easily it should be separable, and these explicit solution are really useful while modelling a system). And substituting v=y/x gives us a separable equation. you can watch this video on khan academy:Homogeneous-equations