What does it mea for an equation to be homogeneous?

[bmed90]

As the title says, what does it mean for an equation to be homogenous?

 

[dentedduck]

Check out the link:

http://mathworld.wolfram.com/HomogeneousOrdinaryDifferentialEquation.html

 

[HallsofIvy]

Unfortunately, there are two quite different uses of the term "homogeneous" in differential equations.

1) As applied to first order equations, an equation of the form $y/dx= f(x,y)$ is "homogeneous" if and only if f(ax, ay)= f(x, y) for any number a. That is the same as saying that f can be thought of as a function of y/x only.

2) As applied to a linear equation of order higher than 1, the equation $a_n(x)d^n/x^n+ a_{n-1}(x)y/dx^{n-1}$$+ \cdot\cdot\cdot+ a_1(x)dy/dx+ a_0(x)y= f(x)$ is homogeneous if and only if f(x)= 0 for all x.

 

[bmed90]

Hey thanks for your replies. I guess I should have been more specific in regards to the fact that I was inquiring about 2nd order diff eqs. @ HallsofIvy YOu answered my question perfectly. Simple and straight to the point. Thanks