What does it mean for an equation to be homogeneous?

[Vitani11]

Homework Statement

I have taken ODE, linear algebra, mechanics, math physics, etc. and we would always go on about how important the homogeneous equation is. To solve for the equation of motion for a harmonic oscillator (for example) we would solve for both the homogeneous and particular solution to get the general solution. I am now in PDE and this is coming up often for the heat equation. Now I would like to know what exactly it means for you to have to find the HG solution and particular solution from a more conceptual standpoint. From looking at the equations it seems to be that the HG equation represents an equation that doesn't have anything affecting it. So for a damped harmonic oscillator this thing would be dampening to 0 or for the heat equation this thing would not be losing any heat and so ∂u(x,t)/∂t = 0 (Keeping boundary conditions and initial conditions simple). I noticed that regardless of whether there is a particular solution there is always a homogeneous solution. Is that just what this means though? So when you find the general solution to an equation you find the solution corresponding to the part of the equation which is not influenced or HG (i.e. it's "natural state") and the solution to the part of the equation which is influenced by parameters or what have you. Is this correct?

Homework Equations

examples:
1. ∂u(x,t)/∂t= ∂2Φ/∂x2 where ∂2Φ/∂x2 = 0
2. d2x/dt2+2βdx/dt+ω2x = 0 for HG
and d2x/dt2+2βdx/dt+ω2x = f(t) for particular

The Attempt at a Solution

None specific

 

[Vitani11]

Also as a side what is λ in the heat equation? I mean you split u(x,t) up and rearrange and set it equal to some constant λ. Are you in effect creating a constant of proportionality which relates the heat flow to change in flux? Last question though - what is the second derivative of heat flux with respect to position? How do you imagine this? My brain (and many others maybe) is too used to imagining things moving with respect to time when thinking of derivatives.

 

[Mark44]

The definition of a homogeneous differential equation is something you should have seen in your ODE class. Assuming that the dependent variable is y and the independent variable is t, a homogeneous differential equation consists of combination of y, y', y'', and possibly higher derivatives on one side of the equation, and 0 on the other.

For example, y'' - 3y' + 2y = 0 is a homogeneous linear differential equation. y'' - 3y' + 2y = e^t is a nonhomogeneous linear differential equation.

I mean you split u(x,t) up and rearrange and set it equal to some constant λ. Are you in effect creating a constant of proportionality which relates the heat flow to change in flux?

Show me an example of what you're talking about. I took a course on PDE many years ago, and for one class of problems we assumed that could separate u(x, t) into one function of x alone and another function of t alone.

My brain (and many others maybe) is too used to imagining things moving with respect to time when thinking of derivatives.

Then you need to stop thinking of derivatives in terms of velocity and accleration, and start thinking of them as rates of change of some quantity (not necessarily distance) with respect to some other quantity (not necessarily time).

 

[Vitani11]

Yes I know what a homogeneous and inhomogeneous diff eq are I just wanted a more conceptual explanation in terms of physics. But anyway you are right, it is hard to think in terms of changes in dimensions other than time for me because of physics courses lol. Anyway, thanks.

 

[Mark44]

In terms of the physics involved, the homogeneous ODE y'' + 3y' + 2y = 0 could represent a mass-spring system with a damper (or and LRC circuit). If y(0) = y₀, and y'(0) = v₀ a physical interpretation of this initial value problem is that the mass is originally at a position y₀ and is moving at a velocity of v₀ when t = 0. Eventually, the system will come to rest.

Here's a nonhomogeneous ODE of the same system: y'' + 3y' + 2y = sin(t), with the same initial conditions as before. The sin(t) term causes the equation to be nonhomogeneous, and is called the "forcing function." Instead of coming to rest, the system will continue oscillating, due to the forcing function.

 

[zwierz]

Now I would like to know what exactly it means for you to have to find the HG solution and particular solution from a more conceptual standpoint.

Conceptually it is a very fundamental and simple (like all fundamental ideas) idea. Let $X,Y$ stand for the vector spaces, say over the field $\mathbb{R}$. And let $A:X\to Y$ be a linear operator. What can we say about an equation $$Ax=b,\quad b\in Y,\quad b\ne 0.\qquad (*)$$ This is a nonhomogeneous equation. Consider a homogeneous equation $$Ax=0\qquad (**).$$

Theorem. Let $\tilde x$ be a solution to equation (*). Then any other solution $x'$ to (*) is presented as follows $x'=\tilde x+w$ where $w$ is a solution to (**).
Indeed, by linearity of the mapping $A$ it is easy to see that $x'-\tilde x$ is a solution to (**)

For example, for equation $y'+y=f(x)$ the operator $A$ is $Ay=y'+y$ and $b=f(x)$