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Determining if Systems are Linear

  1. Jan 16, 2017 #1

    squelch

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    Gold Member

    1. The problem statement, all variables and given/known data

    For each of the following, determine if the system is linear. If not, clearly state why not.
    (a) ##y''(t)+15y'(t)+sin(y(t)))=u(t)##
    (b) ##y''(t)-y'(t)+3y(t)=u'(t)+u(t)##
    (c) ##y'(t)=u(t)## and ##z'(t)=u(t)-z(t)-y(t)##

    2. Relevant equations

    None

    3. The attempt at a solution

    Intuitively, I believe that the ##sin(y(t))## term makes (a) nonlinear, while all of the others can be expressed as linear differential equations, but I wasn't sure. I was hoping for a bit of a sanity check on this. Does the introduction of a trigonmetric function like sin() make these nonlinear all of the time, or is it just that the function is nested, e.g. as ##sin(y(t))##?
     
  2. jcsd
  3. Jan 16, 2017 #2

    Mark44

    Staff: Mentor

    The first equation (a), is nonlinear. A linear differential equation consists of a linear combination of the dependent variable (y(t) in this case) and its derivatives. By "linear combination" I mean a sum of constant multiples of the the dependent variable and its derivatives. Having the sin(y(t)) term makes this equation nonlinear.
     
  4. Jan 16, 2017 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    You are correct: a linear DE is one in which ##y(t)## and all its time-derivatives appear linearly.

    That means that we can have coefficients that are functions of ##t## (linear or nonlinear) and still have a linear DE. So, for example, the equation ##t^2 y''(t) - 2 t y'(t) + t \sin(t) y(t) = 0## is still regarded as a linear differential equation.
     
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