I'm looking to solve an issue I am having with the definition of what makes an ODE (Linear or Nonlinear) Homogeneous. It will also settle a debate we are having involving the definition and example provided in our university's course notes.
It is similar to the questions here and here.
The first link relates to the definition that we have in our DE course notes. The course notes were put together by one of our lecturers, and the definition is based on Ordinary Differential Equations with Boundary-Value Problems by Zill and Cullen.
The equation used as an example is $$c_n \left( \dfrac{d^nx}{dt^n}\right) + c_{n-1}\left( \dfrac{d^{n-1}x}{dt^{n-1}}\right)+\ldots + c_1 \left(\dfrac{dx}{dt}\right)+c_0 x = f(t),$$
and the definition provided is
Definition (Homogeneity of a Differential Equation) A differential equation is homogeneous if for a given solution $x$ to that differential equation, any constant multiple cx is also a solution to that differential equation, otherwise it is non-homogeneous. This leads to the observation that every term in a homogeneous DE contains only the dependent variable or its derivatives.
The notes then conclude that the equation is non-homogeneous because the term $f(t)$ does not contain the dependent variable $x$. If f(t) = 0 then the equation becomes homogeneous.
The same lecturer then states that the equation
$$\left(\dfrac{d^2 x}{d y^2}\right)^5 + x^3 \left(\dfrac{dx}{dy}\right) = x^5$$
Is non-homogeneous as it fails the test provided in the given definition. i.e. $x=cx$ shows the equation is non-homogeneous.
$$c^5\left(\dfrac{d^2 x}{d y^2}\right)^5 + c^4 x^3 \left(\dfrac{dx}{dy}\right) = c^5 x^5$$
and we see that it is non-homogeneous. At the same time however, I believe this contradicts the statement provided
The equation is non-homogeneous because the term $f(t)$ does not contain the dependent variable $x$.
I can rewrite the equation
$$\left(\dfrac{d^2 x}{d y^2}\right)^5 + x^3 \left(\dfrac{dx}{dy}\right) - x^5 = f(y)$$
and since $f(y)=0$ it should therefore be homogeneous.
Simultaneously, I have from the second link that
A linear differential equation is called homogeneous if the following condition is satisfied:
If ϕ(x) is a solution, so is cϕ(x), where c is an arbitrary (non-zero) constant. Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable y must contain y or any derivative of y.
This tells me that the definition given in our course notes is indeed not applicable for any ODE but rather for linear ODES. The lecturer who set the notes however strongly disagrees.
This leads me to question what I know about Homogeneous DEs so I found the following definition given in
Ibragimov A Practical Course in Differential Equations and Mathematical Modeling, §3.1.3 "Homogeneous Equations", p. 93:
An ordinary differential equation of an arbitrary order $$F(x,y,y',…,y^{(n)})=0$$ is said to be homogeneous [in general] if it is invariant under a scaling transformation (dilation) of the independent and dependent variables […]: $$\bar{x}=a^kx,\qquad\bar{y}=a^ly,$$ where $a>0$ is a parameter not identical with 1, and $k$ and $l$ are any fixed real numbers. The invariance means that $$F(\bar{x},\bar{y},\bar{y}',…,\bar{y}^{(n)})=0,$$ where $\bar{y}'=d\bar{y}/d\bar{x}$, etc.
This can also be found here on stackexchange with other definitions here.
This however means that differential equations such as
$$\dfrac{dy}{dx}+y^2 = \dfrac{C}{x^2}$$
are homogeneous as the equation is invariant under
$$\bar{x}=a x \quad and \quad \bar{y} = a^{-1}y$$.
Here $f(x)\ne 0$ yet the ODE is homogeneous.
Is there a 'test' that works for all ODEs linear or nonlinear? Should we split the definition of homogeneous DEs into 2 separate definitions? One for linear and one for nonlinear. Can mathematicians agree on one definition as clearly there are conflicts even between textbooks. Maybe more clarity should be given between homogeneous definitions and when they apply or when they don't.
Any feedback or advice would be great. Preferably if there is a concrete definition that agrees on whether or not the equation in question is or is not homogeneous.
Many Thanks and awaiting feedback
Kendall
EDIT: Not a duplicate of What is a homogeneous Differential Equation?
My question seeks to confirm definitions between both LINEAR and NONLINEAR homogeneous ODEs. The duplicate in question refers only to LINEAR ODEs. My question also seeks to confirm the definition as provided in our course notes as the lecturer INSISTS that definition is applicable to BOTH LINEAR and NONLINEAR ODEs. The duplicate question indicates that definition to only be applicable to LINEAR ODEs but with no sources to confirm that, unlike those as given by Geremia, holds absolutely no credit to its accuracy.
