1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook