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Help toward solving second order non-linear differential equation

  1. Jul 22, 2010 #1

    I have a differential equation of the form
    d2 x
    ---------------- = g/z * x(t)

    Here g and z are constants. So, this is a 2nd order ODE which has a closed form solution.
    In fact, i know the solution for x in terms of cosh and sinh functions.

    In the above differential equation, if z is not constant. That is, id the the eqn becomes
    d2 x
    ---------------- = g/z(t) * x(t)

    Is it still correct to assume x is only a function of time?
    If z(t) is sinusoidal, then this equation is a non-linear 2nd order ODE. Am i right?

    Please point me to a standard textbook which might help me solve these kind of non-linear diff.equations? Any suggestions/pointers are welcome. Thank You in advance.

  2. jcsd
  3. Jul 22, 2010 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    It's still a linear differential equation. If x and y are both solutions

    [tex] \frac{d^2(x+y)}{dt^2}=\frac{d^2x}{dt^2}+\frac{d^2y}{dt^2}=\frac{g}{z(t)}x + \frac{g}{z(t)}y = \frac{g}{z(t)}(x+y)[/tex]

    You can also show that scalar multiples are solutions
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