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Solving the differential equation of an object oscillating in water.

  1. Apr 14, 2014 #1
    I have a differential equation to solve below on the motion of an object oscillating in water with a restoring force equal to -Aρgx and a damping force equal to -kv^2.
    ma+kv^2+Aρgx=0
    K, A, ρ and g are constants and I need to solve the equation for x. a (acceleration). v (velocity). x (displacement from the equilibrium position).
    I need a bit of help on this one because I dont know whether it would need a substitution to eliminate v^2.
     
  2. jcsd
  3. Apr 14, 2014 #2
    Your differential equation is of the form

    [itex]x'' = f(x,x')[/itex]

    where primes indicate derivatives wrt to time. When ever the independent variable (in this case time) does not appear explicitly in f, then try the substitution

    [itex]x' = z[/itex].

    Using the chain rule you can show that
    [itex]x'' = z\frac{dz}{dx} = \frac{1}{2} \frac{d\left(z^2\right)}{dx}[/itex].

    Thus the substation converts a second order nonlinear equation into a first order nonlinear equation.

    In your case, you can solve the resulting equation by using an integrating factor.
     
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