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Linear and non-linear differential equations

  1. Aug 16, 2012 #1
    I am not sure I understand Linear and non-linear differential equations properly so I will ask some question which someone will hopefully answer.

    Is dx/dt = x + 2 a linear differential equation? If so does this mean that the rate of change of is constant?

    Is dx/dt = x^2 + 4x a non linear differential equation?

    Any answers would be appreciated :smile:
     
  2. jcsd
  3. Aug 17, 2012 #2
    Linear ordinary differential equations are statements of the form:

    linear combination of x(t) and its time derivatives = f(t)

    The independent variable doesn't have to be called t, but it's a nice convention.

    The first equation can be written:
    ## -x(t) + \frac{d}{dt}x(t) = 2##
    The left side is a linear combination of ##x## and its first time derivative, and the right hand side is a (boring, constant) function of time. So it is a linear ODE.

    In the second equation, the ##x^2## term prevents us from writing the ODE in the form linear combination of x and its derivatives = f(t). So it is nonlinear. For nonlinear ODEs, the superposition principle isn't guaranteed to work, and some other bad behaviors are allowed that would be impossible for linear systems. For example, I think your second equation grows hyperbolically: it blows up to ∞ in finite time.
     
  4. Aug 17, 2012 #3
    First, it is much easier for beginners to consider the dependencies of [itex]t[/itex] in the functions. So in your example [itex]x[/itex] is [itex]x(t)[/itex].

    Try to write the differential equations in the following form:

    [itex]x’(t) = A(t)x(t) + g(t)[/itex]

    If this is possible it is a linear differential equation.

    In your first example: What is [itex]A(t)[/itex] and what is [itex]g(t)[/itex]?

    In the second example: Why is the above form not possible?
     
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