Differential equation linear input/output

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SUMMARY

The discussion centers on the linearity of the differential equation my'' + cy' + ky = ku + cu', which models a physical system involving mass (m), damping (c), and spring (k). The key conclusion is that the output y(t) scales linearly with the input u(t) due to the properties of linear differential equations, where if u(t) is multiplied by a constant C, the output y(t) also scales by the same factor. This relationship holds true as both u(t) and y(t) satisfy the equation for all values of t, confirming the linear input/output nature of the system.

PREREQUISITES
  • Understanding of linear differential equations
  • Familiarity with the concepts of homogeneous and particular solutions
  • Knowledge of physical systems modeling (mass, damping, spring)
  • Basic principles of linearity in systems theory
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Homework Statement



Given the equation my'' + cy' + ky = ku + cu' explain why the size of the output scales linearly with the size of the input. In your argument, do NOT actually try to find the solution(s) to the equation, rather use the definition of what it means to be a solution to a DE in your explanation.

Homework Equations



my''(t) + cy'(t) + ky(t) = ku(t) + cu'(t)

This equation models a physical system where m is a mass, c is a damper, and k is a spring. y is the output and u is the input.

The Attempt at a Solution



I'm probably reading too much into this, given the equation is not very simple but the explanation supposedly should be pretty easy.

I know the solution is made up of the homogeneous and particular parts. Since the particular solution is the response to input, I'm guessing the explanation has something to do with that. The solution y(t) should also satisfy the equality in the equation for all values of t in order to be a solution. Must y(t) = u(t) and thus increase u(t) by a factor of some constant will thus affect the output?

I've done searches on linear input/output systems, but most sources give lengthy mathematical discussions about why they are linear. Could someone steer me in the right direction please?
 
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Well, if u(t) and y(t) solve that equation, and C is some constant, then don't C*u(t) and C*y(t) also solve it? This wouldn't be true if the equation were something like y'(t)^2=u(t), right? I think it's as simple as that.
 

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