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Confused on the directions! Airy's equation, series solutions! weee!

  1. Mar 1, 2006 #1
    Find a series solution for Airy's equation about x=-1, what does this about x=-1 mean?

    Here is Airy's equation:
    y''+p(t) y'+q(t) y=0.

    THe professor doesn't give any 2nd order Differential equation, just the directions of:
    Fridays Homework problem

    Is to find a series solution for Airy's equation about x=-1

    Can i use this as my equation?
    y''-t y=0
    and then:

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    These equations are known as the "recurrence relations" of the differential equations. The recurrence relations permit us to compute all coefficients in terms of a0 and a1.

    We already know from the 0th recurrence relation that a2=0. Let's compute a3 by reading off the recurrence relation for n=1:

    [​IMG]

    [​IMG]



    But what is this thing all about x=-1?

    Thanks!
     
    Last edited: Mar 1, 2006
  2. jcsd
  3. Mar 2, 2006 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Either you or your teacher is really confused!

    First of all, y"+ p(t)y'+ q(t)y= 0 is NOT "Airy's equation". That is the general form of a linear homogeneous equation with variable coefficients.
    Airy's equation, specifically y"- ty= 0, is a special case of that.

    Second "solve y"- ty= 0 with a series solution about x= -1" makes no sense because there is no "x" in the equation! You must have either
    "Solve y"- ty= 0 with a series solution about t= -1" (y" is understood to be differentiation with respect to t) or "Solve y"- xy= 0 with a series solution about x=-1" (y" is understood to be differentiation with respect to x).

    Use [itex]\Sigma a_n (t+ 1)^n[/itex] rather than [itex]\Sigma a_n t^n[/itex]. If, for example, you were given initial conditions at t= -1 rather than t= 0, that form would make it easier to apply them.
     
  4. Mar 2, 2006 #3
    Ivey if i already have the general solution to aries equation, can i just make alitle modifcation to it for x= -1?
    [​IMG]

    [​IMG]
     
  5. Mar 2, 2006 #4
    Ahhh n/m, i c i can't just do that, i will try it with (t+1) like u said! thanks
     
  6. Mar 2, 2006 #5
    Did i do this right?>

    Ivey and everyone, i think i did this right, its a long problem, but I wrote what i was going to do next so I think its easy to follow, did I do this correctly? This is the first type of problem I did like this so i'm making sure i didn't leave out somthing!

    [​IMG]
    [​IMG]

    Thanks!
     
  7. Mar 2, 2006 #6
    I went over it again and i caught a small error i think:
    i had
    a2 + ao + ....
    should have been:
    2a2 + ao
    wewt.
     
  8. Mar 3, 2006 #7
    I presented it today and it was correct! weee!
     
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