Linear-Homogeneous vs Homogeneous ODEs?

Question

Currently in my third week of my first ODEs class and I've already encountered something I'm struggling with. My second homework assignment requires me to classify and solve some ODEs. He gave us four classifications: separable, linear, homogeneous, and Bernoulli.

However, when reading through my text, I noticed there was a split between homogeneous and non-homogeneous linear equations, in which a homogeneous linear ODE has the form $y\prime + p(t)y = f(t)y^n$ where $f(t) \equiv 0$. Later on, the author discusses homogeneous ODEs again, but this time saying an ODE is homogeneous if it can be written as $y\prime = f(\frac{t}{y})$ for some function $f$.

So I have two questions: (1) Are those "homogeneous" descriptions the same thing? Is the whole $f(t) \equiv 0$ a corollary to the definition of homogeneous and specific only to linear ODEs? and (2) Is it possible for an ODE to be categorized by multiple descriptors (i.e. Bernoulli and homogeneous, linear and separable)?

Answer 1

Unfortunately, the word "homogeneous" has two different meanings in the context of ODE's.

1. A linear equation is said to be homogeneous if there is no nonzero term not involving the dependent variable. Thus $y'' + p(t) y' + q(t) y = 0$ is homogeneous, but $y'' + p(t) y' + q(t) y = r(t)$ is not (unless $r$ happens to be $0$).
2. A first-order equation of the form $$\dfrac{dy}{dt} = F\left(\frac{y}{t}\right)$$ is said to be homogeneous.

These are completely different concepts, unrelated to each other.

Answer 2

(1) Yes and no :) For this stage it it better to distinguish between homogeneoud first order equations $$ \dot y=f(y/t) $$ and linear homogeneous equations $$ \dot y+p(t)y=0. $$

(2) Of course. For example, the linear homogeneous equation is separable.