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Recognizing linear ordinary differential equations

  1. Sep 8, 2011 #1
    1. The problem statement, all variables and given/known data

    Is the following equation a linear ODE?
    [itex]\frac{d^{2}R}{dt^{2}}=-\frac{k}{R^{2}}[/itex] where k is a constant

    2. Relevant equations
    A linear ordinary differential equation can be written in the following form:
    [itex]a_{n}\left ( x \right )\frac{{d}^{n}y}{{d}x^{n}}+a_{n-1}\left ( x \right )\frac{{d}^{n-1}y}{{d}x^{n-1}}+...+a_{1}\left ( x \right )\frac{{d}y}{{d}x}+a_{0}\left ( x \right )y=g\left ( x \right )[/itex]


    3. The attempt at a solution
    Well, the correct answer in the textbook is: not a linear ODE. But there is something I don't really understand. If we multiply both sides by [itex]R^{2}[/itex] then we have:
    [itex]R^{2}\frac{d^{2}R}{dt^{2}}=-k[/itex]

    The right hand side is of the form [itex]g\left ( x \right )=-k[/itex], so this is good. Also, the [itex]\frac{{d}^{2}R}{{d}t^{2}}[/itex] term is power of 1, which is also good. The problem relies in the [itex]a_{n}\left ( x \right )[/itex] term. The coefficient in front of the derivative must at most depend on the independant variable, in this case, t.

    We have [itex]R^{2}[/itex]. Well, in my opinion, we can't know whether it is linear or not because we don't know how R is explicitely defined. R could be a function of t in the following form: [itex]R=F\left ( t \right )[/itex]. If this is the case, then [itex]R^{2}[/itex] could be substitued by [itex]\left (F\left ( t \right ) \right )^{2}[/itex]. Then we would clearly see that [itex]R^{2}[/itex] depend only on the variable t and we could conclude that the given equation is linear.

    Given my arguments, I don't understand why we can already tell that the equation is non-linear. Thank you very much!!
     
  2. jcsd
  3. Sep 8, 2011 #2

    micromass

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    Well, even if [itex]R=F(t)[/itex], that would not make it linear. The problem is that F is an unknown function. That is: we don't know (a priori) the exact form of F.

    The idea is that [itex]\alpha_i[/itex] are known functions that depend on t. But R is not a known function. Thus it is a non-linear equation.
     
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