"Rationale" for Homogeneous vs. Nonhomogeneous Differential Equations?

[Martyn Arthur]

TL;DR Summary

Trying to understand the "rationale" of the result of the 2 types and in particular the relevance of zero

Hi; I am missing something. I can follow the technicality of a homogenous linear equation has all coefficients of zero and the "contra" for non homogenous equations. I just can't figure out the relevance of the consequences of outcome. If I am not being clear maybe I can be guided as to how better make my confusion about the subject clearer. Thanks as always Martyn Arthur

[Post edited by a Mentor to remove a forum glitch]

 

[Mark44]

TL;DR Summary: Trying to understand the "rationale" of the result of the 2 types and in particular the relevance of zero

Hi; I am missing something. I can follow the technicality of a homogenous linear equation has all coefficients of zero and the "contra" for non homogenous equations.

I don't understand the part about all coefficients being zero.

Unfortunately, the term "homogeneous" as a descriptor of differential equations has two different meanings. This wiki, https://en.wikipedia.org/wiki/Homogeneous_differential_equation, discusses both types.

I'm assuming that you're asking about a homogeneous linear differential equation of a sort like this example:
y'' + y' - 6y = 0
Here, y(x) is an unknown function, and the goal is to determine the solutions to this equation. As it turns out, two solutions are ##y(x) = e^{-3x}## and ##y(x) = e^{2x}##. In fact, there are an infinite number of solutions. Any scalar multiple of either of the above functions is also a solution. The general solution lists all possible solutions - ##y(x) = c_1e^{-3x} + c_2e^{2x}##, with ##c_1## and ##c_2## being arbitrary constants. If some initial conditions are given, say, y(0) = A and y'(0) = B, then a unique solution can be determined.

An example of a nonhomogeneous differential equation is y'' + y' - 6y = 3. The constant term on the right side is what makes it nonhomogeneous.

If you can refine your question, maybe we can answer other questions.

 

[renormalize]

Unfortunately, the term "homogeneous" as a descriptor of differential equations has two different meanings. This wiki, https://en.wikipedia.org/wiki/Homog...ia.org/wiki/Homogeneous_differential_equation, discusses both types.

The posted wiki link is incorrect, it should be:
https://en.wikipedia.org/wiki/Homogeneous_differential_equation

 

[vela]

Have you studied linear algebra yet?

 

[Mark44]

The posted wiki link is incorrect, it should be:
https://en.wikipedia.org/wiki/Homogeneous_differential_equation

Thanks! Somehow, I got the URL in there twice. I've fixed it now, as well as a typo in one of the LaTeX expressions.

 

[DaveE]

In my world of EE (analog circuits, controls, feedback, etc.) we work with both for different reasons. If you want to characterize a system in a general sense, we will use the homogeneous forms or simple driving functions like δ or step response. If we have a specific requirement that includes a driving function then we will solve the transient response which includes the details of the input(s).

Frankly, in design, we mostly care about the former, since we don't usually know the details of what the system will experience out in the real world. So things like stability analysis and impulse response are all about the homogeneous cases. For example, you want to design a car suspension to work well with all sorts of bumps, not one specific pothole.

Tools like Fourier and Laplace Transforms will help make the connection from a good general design to a good response to a specific input.

 

[vela]

I just can't figure out the relevance of the consequences of outcome.

Can you explain what you mean by the phrase consequences of outcome? It's extraordinarily vague. What consequences of what outcomes? Perhaps a specific example can help to illustrate what you're asking about.

 

[Martyn Arthur]

Thanks for the patience!
I have ADHD and it happens that stuff that is clear generally sometimes gets muddled.
I just need a basic understanding, maybe I am muddling the wholee question.
Could we maybe please get back to an absolute context leaving aside calculations and equations but having regard to just the absolute.
What is the difference between homogenous and inhomogeneous equations.
Is it "just" that homogenous functions represent, when "graphed" a continuous slope?
Thanks
Martyn

 

[vela]

What is the difference between homogenous and inhomogeneous equations.

Did you read @Mark44's post above? He explained the difference there. Or do you have a textbook? It's surely explained there.

 

[Martyn Arthur]

I am grateful to this comunity for its ongoing patience and help.
Thank you.
I have now the answer to my question.
I am posting a separate thread about Hydrogen; I know my abstract way of looking at things, undefined, will meet with the same patience.
Thanks
Martyn Arthur

 

[WWGD]

There's also the fact that the solution set to such system, if/when it exists, is a vector space. Finite dimensional too, most of the time, iirc. That's a nice amount, type of structure to have for your solution set.