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

 

[blue_raver22]

A homogeneous equation is one where all the terms have the same degree of variables. This means that each term in the equation has the same number of variables raised to the same powers. In other words, the equation is balanced in terms of the variables present. This allows for simpler manipulation and solution of the equation, as the variables are all of the same type and can be treated equally. Homogeneous equations also have the property of being invariant to scaling, meaning that if all the variables are multiplied by a constant, the equation still holds true. This property makes homogeneous equations useful in many areas of science, particularly in physics and chemistry, where they often represent physical laws and relationships between variables. In summary, a homogeneous equation is one that is balanced and invariant to scaling, making it easier to work with and applicable to a wide range of scientific problems.