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I'm currently a third year undergrad electrical engineering student studying linear control system and very much confused regarding the term "linearity".

  1. Does the term "A linear ODE" necessarily mean that the system it is describing has to be a "linear system"?
  2. Given a differential equation, can one directly test the linearity (Homogeneity and Superposition) of the equation without first finding its solution?

If you can explain the concepts in a more simpler manner instead of rigorous math, that would be appreciated. But if it's necessary to understand the math behind it, I'll try my best. Thanks!

C.Suwan
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  1. Yes. In practice it may only be an approximation, and it is almost certainly only linear across a defined input/output range (e.g. consider what happens when an opamp saturates).

  2. Well, if you put a sine wave into the system, and it only produces a sine wave at the same frequency at the output, then it is linear. If you have the equation it is trivial to determine linearity (hint, check there are no non-linear operators used on the independent variable).

Jon
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Linear means different things in different contexts:

In controls or ODE's it means that the input is proportional to the output. It also means that you can apply laplace and the functions will add in the frequency space which greatly simplifies calculations.

Given a differential equation, can one directly test the linearity (Homogeneity and Superposition) of the equation without first finding its solution?

Yes, you don't even need to see the function, just put in a step input or impulse input and then observe the output, if the outputs are proportional then you have a good case that it's linear. (I used this property on a physical system to find it's ODE or open loop plant function)

Voltage Spike
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1.linear constant cofficient differential equations are subset of linear system , so you can definitely say that a LCCDE represent a linear system

2.yes , you can check directly linearity by applying conditions of homogeneity and superposition in input output variable of differential equation (without solving it)

user215805
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