[Math] Can a Homogeneous Differential Equation be Nonlinear

Question

First of all, I am not asking about $v=y/x$ transformation kinda homogeneous.

Can we say this nonlinear differential equation is homogeneous $$y^{\prime}=ty^2.$$

Here there is no term without $y$, this is okay. But this is nonlinear equation.

I saw here some discussions and some people said it is just for linear equations. I want to give two links:

Dummies Series (assumes it can be nonlinear) (I also see some universities documents too).

Wolfram Math World (assumes it has to be linear)

Which one is the definition:

1) No term without $y$,

2) Linear and No term without $y$.

3) $y$ is a solution, then $\lambda y$ is solution for all $\lambda \in \mathbb R$.

Considering the references, I will use the 3rd one. This option is also meaningful.

Answer 1

The answer to the title-question is: Yes, it can. Here are two references.

The paper On the second order homogeneous quadratic differential equation by Roger Chalkley considers homogeneous quadratic differential equations of the form \begin{align*} Q(y)\equiv a\left(y^{\prime\prime}\right)^2+by^{\prime\prime}y^{\prime}+cy^{\prime\prime}y+d\left(y^{\prime}\right)^2 +ey^\prime y+fy^2=0\tag{1} \end{align*}

from which we conclude linearity and homogeneity are two concepts which do not exclude each other.

Note: The Math world page does not include the term linear in the definition of Homogeneous Ordinary Differential Equation. It rather uses a linear differential equation as an easy to follow example for a homogeneous ordinary differential equation.

Another example is the paper

Classification and Analysis of Two-Dimensional Real Homogeneous Quadratic Differential Equation Systems by Tsutomu Date.