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Exponential Forcing Differential Equation

  1. Apr 9, 2017 #1
    1. The problem statement, all variables and given/known data
    Solve ## \frac{d^2y}{dt^2} + \omega^2y = 2te^{-t}##
    and find the amplitude of the resulting oscillation when ##t \rightarrow \infty ## given ##y=dy/dt=0## at ##t=0##.

    2. Relevant equations


    3. The attempt at a solution
    I have found the homogenious solution to be:
    ##y_h = A\cos\omega t + B\sin\omega t ##
    where A and B are constants.
    When looking for the particular integral I tried the obvious choice ##y_p = Cte^{-t}##. However, unless I have done a misstake this yields an equation system:
    ##(1+\omega^2)Cte^{-t} = 2te^{-t}##
    ##-2e^{-t}=0##
    which lacks solution. Any ideas what more I should try?

    I can see that as ##t\rightarrow \infty## the forcing term will tend to ##0## and hence the final amplitude should be ##\sqrt{A^2+B^2}##, but I would like to find the solution to the equation..

    Many Thanks! :)
     
  2. jcsd
  3. Apr 9, 2017 #2
    Don't bother, I realised that ##y_p = Cte^{-t}+De^{-t}## works! :)
     
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