Solution for a second-order differential equation

  • Thread starter chaksome
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  • #1
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Homework Statement:
Second order ODE
Relevant Equations:
x‘’(t)+4x’(t)+8x(t) = f(t)
I wish to know if there is a method to work out x(t).
[No matter which form f(t) is]
Thank you~
 

Answers and Replies

  • #2
BvU
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You know PF: what's your attempt ? Know about Laplace transforms ?
 
  • #3
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You know PF: what's your attempt ? Know about Laplace transforms ?
Oh, I haven‘t learnt that^v^
Do you mean this chapter will help me understand this problem?
I can try to learn it~
 
  • #4
BvU
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It gives you a formal, general method to find a solution, yes.
 
  • #5
HallsofIvy
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First, the "associated homogenous equation" is x''+ 8x'+ 4x= 0 which has "characteristic equation" r^2+ 8r+ 4= 0. Solve that by "completing the square": r^2+ 8r+ 16- 16+ 4= (r+ 4)^2- 12= 0 so (r+ 4)^2= 12, [itex]r= -4\pm2\sqrt{3}[/itex]. The general solution to the "associate homogeneous equation" is [itex]e^{-4x}\left(Ae^{2x\sqrt{3}}+ Be^{-2x\sqrt{3}}\right)[/itex].

Now use "variation of parameters". We look for a solution to the entire equation of the form [itex]y(x)= e^{-4x}\left(u(x)e^{2x\sqrt{3}}+ v(x)e^{-2x\sqrt{3}}\right)[/itex]. That is, we let the "parameters", A and B, vary. Surely That method is given in your text book. You can also read about it at http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx.

The calculations are tedious but doable! The result will be an integral in f(x) that you add to the general solution to the homogeneous equation.
 

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