Exponential Forcing Differential Equation

  • #1
177
11

Homework Statement


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##.

Homework Equations




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! :)
 

Answers and Replies

  • #2
177
11
Don't bother, I realised that ##y_p = Cte^{-t}+De^{-t}## works! :)
 

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