I wonder where you saw that definition of "homogeneous". There are, in fact, two types of "homogenous" defined for differential equations. When we are talking specifically about first order equations, we say that f(x,y)dy+ g(x,y)dx= 0 is "homogenous" if and only if [tex]\frac{f(\lambda x, \lambda y)}{g(\lambda x, \lambda y)}= \frac{f(x, y)}{g(x, y)}[/itex] that is the same as saying that if we were to write this as dy/dx= F(x,y), F would actually depend only on y/x, it cannot be a general function of x, y and we can simplify the equation by substituting u= y/x. This equation, while first order, is clearly not "homogeneous" in that sense.

There is also a definition of "homogeneous" forlineardifferential equations of any order. This equation is not linear so that does not apply. This equation is not "homogeneous' in either sense.

(And the "unknown" is thedependentvariable, not independent variable.)