Classifying a differential equation

Question

How do I classify the following differential equation? In particular, is this differential equation "homogeneous?"

$$(x^3+3y^2)dx-2xydy=0$$

Solving it is not the problem, but I don't know how to recognize it.

Answer 1

A differential equation is called homogeneous if the sum of powers of variables in each term is constant i.e. each term is in dimensional balance.

As given differential equation: $(x^3+3y^2)dx-2xydy=0 \iff x^3+3y^2-2xy\frac{dy}{dx}=0$

It is clear that sum of powers of (of $x$ & $y$) first (left most) term is $3$ while second & third terms have sum of powers equal to $2$ i.e. sum of powers of each term is not constant (equal).

Hence, the given differential equation is not homogeneous.

Answer 2

first order nonlinear ordinary

first order: because you have $\frac{dy}{dx}=y'$ and not $y''$ or $y'''$

nonlinear: because you have $y^2$

Answer 3

From wikipedia:

Definition. A linear differential equation is called homogeneous if the following condition is satisfied: If $\phi(x)$ is a solution, so is $c \phi(x)$, where $c$ is an arbitrary (non-zero) constant.