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Is the system linear or nonlinear

  1. Sep 8, 2016 #1
    1. The problem statement, all variables and given/known data
    3y(t)+2=x(t)

    2. Relevant equations

    k1y1(t) + k2y2(t) + 2(k1+k2) = k1x1(t)+k2x2(t)

    3. The attempt at a solution

    I know the system is non linear but I cannot explain why. It has something to do with 2(k1+k2) but I am unsure.
     
  2. jcsd
  3. Sep 8, 2016 #2

    berkeman

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    Staff: Mentor

    Why do you say you know it's non-linear?
     
    Last edited: Sep 8, 2016
  4. Sep 8, 2016 #3

    Mark44

    Staff: Mentor

    How are you defining the term "linear"? In some contexts, this term implies that the graph of the relationship is a straight line. In the context of linear transformations, the conditions for linearity are that ##L(x_1 + x_2) = L(x_1) + L(x_2)## and that ##L(cx) = cL(x)##.

    You put this equation in the Relevant equations section. How is it relevant to this problem?
     
  5. Sep 9, 2016 #4
    it is both the properties you have described but I have combined them into one equation.

    also it was marked as nonlinear in the solution manual
     
  6. Sep 9, 2016 #5

    Mark44

    Staff: Mentor

    You could write the given equation as ##y(t) = \frac 1 3 x(t) - \frac 2 3##
    Now check the two properties separately.
    1) Is ##y(t_1 + t_2) = y(t_1) + y(t_2)##?
    2) Is ##y(k \cdot t_1) = k \cdot y(t_1)##?

    If both of the above are true for all values of t, the relationship is linear; otherwise, it's nonlinear.
     
  7. Sep 9, 2016 #6

    Ray Vickson

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    Science Advisor
    Homework Helper

    I think it is a bit ambiguous.

    If your equation ##3y(t)+2 = x(t)## describes a "dynamical system", then it is not linear because if ##(x_1(t),y_(t))## and ##(x_2(t),y_2(t))## are two solutions, the pair ##(x_1(t)+x_2(t),y_1(t)+y_2(t))## is not a solution, nor is ##(cx_1(t),cy_2(t))## for a constant ##c \neq 1##.

    On the other hand the equation ##x - 3y = 2## is what would normally be called "linear equation"---meaning that the left-hand-side in ##f(x,y) = c## is a linear function of ##x,y##. Perhaps that is a bit of abuse of language, but it is nevertheless standard usage in describing equations. We do something similar when we call a differential equation such as ##dy/dx + 2 y = x^2## a (non-homogeneous) linear differential equation!
     
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