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General solution to inhomogeneous second order equation

  • Thread starter NT123
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  • #1
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Homework Statement

I need to find the solution to x'' + cx' = f(t), for a general f.



Homework Equations





The Attempt at a Solution



Obviously first I solve the homogeneous part to give me A + B*exp(-ct). I also know that the particular solution is written as (1/c)int((1-exp(c(s-t))f(s))ds where the integral is between 0 and t. However I am not sure why this is so, any help would be much appreciated.
 

Answers and Replies

  • #2
gabbagabbahey
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Have you learned about Fourier Transforms yet? If so, just transform both sides of the DE, solve the resulting algebraic equation for [itex]\tilde(x)(t)[/itex] and then take the inverse Fourier Transform.
 
  • #3
HallsofIvy
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Oh, dear! Using "Fourer Series" for this is like using a shotgun to kill a fly!

Let v= x' and your differential equation becomes v'+ cv= f(t). That's a linear equation with "integrating factor" [itex]e^{ct}[/itex]. That is,
[tex]\frac{d(e^{ct}v)}{dx}= e^{ct}v'+ ce^{ct}v= e^{ct}f(t)[/tex]

Integrating both sides,
[tex]e^{ct}v= \int_{t_0}^t e^{cs}f(x)ds+ C[/tex]

From that,
[tex]v= x'= Ce^{-ct}+ e^{-ct}\int_{t_0}^t e^{cs}f(s)ds[/tex]

Now, integrate again:
[tex]x(t)= C_1 e^{-ct}+ \int_{t_0}^t\left(e^{-cu}\int_{t_0}^u e^{cs}f(s)ds\right)du+ C_2[/tex]
 

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