Nonhomogeneous: Undetermined coefficients

  • #1
(d^2x/dt^2)+(w^2)x=Fsin(wt), x(0)=0,x'(0)=0

Hope that's readable. First part is second derivative of x with respect to t. w is a constant and F is a constant. I need to find a solution to this using method of undetermined coeffecients and I'm confused with all the different variables. Anyone get me started at least?
 

Answers and Replies

  • #2
Pyrrhus
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Well, first off start by solving the homogenous equation to find the fundamental solution.

[tex] \ddot{x} + \omega^{2}x = 0 [/tex]

After that try a Particular solution of the type

[tex] y_{p} = A x \sin(\omega t) + B x\cos(\omega t) [/tex]

Remember that if the fundamental solution has already sin and cos, you will need to try a xsin and xcos, like this case.
 
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  • #3
I got my homogenous equation x''+(w^2)x=0 but I can't find my roots with that w^2 in there.
 
  • #4
Pyrrhus
Homework Helper
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What seems to be the problem? Show me your work.
 
  • #5
Pyrrhus
Homework Helper
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Here, i will start you off

[tex] \ddot{x} + \omega^{2}x = 0 [/tex]

we assume a as a solution

[tex] x(t) = e^{rt} [/tex]

So we substitute in our ODE

[tex] r^{2}e^{rt} + \omega^{2}e^{rt} = 0 [/tex]

so

[tex] e^{rt}(r^{2} + \omega^{2}) = 0 [/tex]

because [itex] e^{rt} [/itex] cannot be equal to 0

[tex] r^{2} + \omega^{2} = 0 [/tex]

which ends up as

[tex] r = \pm \omega i [/tex]
 
Last edited:
  • #6
I figured it out, thanks a lot for your help, I was just being dumb.
 

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