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Homework Help: KILLER 2nd ODE (inhomogeneous) XD

  1. Oct 9, 2012 #1
    Ok, here goes:

    1. The problem statement, all variables and given/known data

    So I've come across this 2nd ODE which I am to "solve ... for a general solution":

    d^2y / dx^2 - dy/dx + y = cos(x) - sin(x) :tongue2:

    and then evaluate the "particular solution" using the boundary conditions y=L when x=0 (also, dy/dx = 0)

    3. The attempt at a solution

    I can't type out the whole of my working because its really long and would be impossible to follow so I'll try and sum up what I've got:

    1) found the general solution of the * equivalent* homogeneous equation... which came out with imaginary values. I then converted this into trigonometric form (as opposed to using imaginary exponentials) which is in the form of:

    exp(1/2 x) ( C sin((sqrt(3)/2)x) + D cos("") ) :grumpy:

    2) I then guessed at the particular solution which I'm thinking looks like:

    (a-b)( cos(x) - sin (x) ) :uhh:

    3) Added ^^these^^ together to get the "General Solution" (y=...) of the original equation. Which looks something like (but with more coefficients and stuff:

    e^... (cos + sin) + ( cos - sin ) {you get the idea} :bugeye:

    4) Trying to evaluate this however lead me to some nasty unsolvable simultaneous equations... :yuck:

    any advice? can you spot my mistake? do I actually just need to solve "2)" to get the answer, I'm confused now...

    this is driving me absolutely crazy.
  2. jcsd
  3. Oct 9, 2012 #2
    The particular solution should be of the form:
    Y_p(x) = A \, \cos x + B \, \sin x

    Determine A and B by plugging this in the ODE. Tell us what you get.
  4. Oct 9, 2012 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    (1) Use a Green's function method http://www.math.umn.edu/~olver/pd_/gf.pdf [Broken] ; or
    (2) Use Variation of Parameters http://en.wikipedia.org/wiki/Variation_of_parameters

    Both methods are standard, but (2) is probably better known.

    Last edited by a moderator: May 6, 2017
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