Why is the right side called the input?

  • Thread starter maxbashi
  • Start date
  • Tags
    Input
In summary: The input in this case would be F_{ext}(t) and the output is x(t).In summary, the right side of the differential equation is often referred to as the "input" because it represents an external force or signal acting on the system being modeled. This can be seen in the solution where the function on the right side produces both a transient and steady-state solution, driving the behavior of the system. The terminology comes from the specific application of the ODE, such as in modeling the dynamics of a physical system.
  • #1
maxbashi
18
0
Why is the right side called the "input?"

I'm looking at linear, 1st order ODEs, like

y' + p(t)x = q(t)

The notes I'm looking at are calling q(t) the "input" of the system, but I'm not sure why. I understand how to solve the equations, but I must be looking at it differently or something. To quote the notes:

The left hand side represents the SYSTEM.
The right hand side represents an outside influence on the system: it's a "signal," the "input signal." A "signal" is just a function of time.
The system responds to the input signal and yields the function x(t), "output signal."

I guess I'm just not intuitively understanding the way the author is looking at ODEs. Any help?
 
Physics news on Phys.org
  • #2


You have too many variables in your differential equation. It almost certainly should be
y' + p(t) y = q(t).

The related homogeneous equation is y' + p(t)y = 0. By separating variables, the solution can be found to be
[tex]|y| = Ae^{-\int p(t) dt} [/tex]

If you are given the initial condition, y(t0) = y0, you can determine A. The function p(t) will determine the behavior of this solution. If this differential equation is modeling a physical process, the typical long-term behavior of this solution is that y(t) --> 0 as t --> infinity.

For the nonhomogeneous equation, y' + p(t)y = q(t), the function on the right side produces a different solution with two parts: the transient solution (which dies out in time), and the steady-state solution. In a sense, q(t) is "driving" the system and can be thought of as acting as an input to the system.
 
  • #3


Let me add that the terminology comes not from the theory of the ODE itself but from some specific typical application. Presumably the ODE is from a model for the dynamics of some physical system. Then the q(t) probably represents an external (time dependent) force.

For example, a mass-spring system has force proportional to position:
[tex] F = -kx[/tex]
So from Newton's laws F=ma:
[tex] m \ddot{x}= -kx + F_{ext}[/tex]
or
[tex]m\ddot{x}+ kx = F_{ext}(t)[/tex]
 

1. Why is the right side called the input?

The right side is called the input because it is where information or energy is received into a system. In scientific experiments and processes, the right side is where the initial data or materials are introduced, hence the term "input".

2. Is the right side always considered the input?

In most cases, yes, the right side is considered the input. However, this can vary depending on the context. For example, in electrical circuits, the left side is often considered the input. It ultimately depends on the specific system or process being studied.

3. What happens if the right side is not used as the input?

If the right side is not used as the input, it may affect the outcome of the process or experiment. The input is typically where the initial conditions are set, so changing this can alter the results. It is important to carefully consider which side is designated as the input in any scientific study.

4. Can the right side also be considered the output?

In some cases, the right side can also be considered the output. This may occur in systems where information or energy is transferred in both directions, such as in feedback loops. However, the right side is more commonly associated with the input, as it is where the initial information or energy enters the system.

5. Why is it important to distinguish between the right and left sides as input and output?

Distinguishing between the right and left sides as input and output is important because it helps clarify the flow of information or energy within a system. It also allows for proper analysis and understanding of the system's behavior and potential changes. Without this distinction, it may be difficult to accurately interpret and predict the outcomes of scientific processes.

Similar threads

  • Differential Equations
Replies
3
Views
1K
  • Differential Equations
Replies
2
Views
972
  • Differential Equations
Replies
1
Views
624
  • Computing and Technology
Replies
3
Views
771
  • Differential Equations
Replies
7
Views
2K
  • Differential Equations
Replies
1
Views
717
  • Differential Equations
Replies
8
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
10
Views
785
  • Engineering and Comp Sci Homework Help
Replies
3
Views
839
Replies
5
Views
4K
Back
Top