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Series solution about a regular singular point (x=0) of xy''-xy'-y=0

  1. May 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the indicial equation and find 2 independent series solutions for the DE:
    xy''-xy'-y=0 about the regular singular point x=0


    2. Relevant equations
    y=Ʃ(0→∞) Cnxn+r
    y'=Ʃ(0→∞) Cn(n+r)xn+r-1
    y''=Ʃ(0→∞) Cn(n+r)(n+r-1)xn+r-2


    3. The attempt at a solution
    Finding the indicial eq.
    Stan. form y''-y'-(1/x)y=0

    p(x)=x*(-1)=-x
    q(x)=x2*(-1/x)=-x

    Making ao and bo both zero for

    r(r-1)+aor+bo=0

    so r=0,1

    Solving for the equation I finish with (I'll skip a few steps, confident in this portion)

    Cor(r-1)xr-1+Ʃ(0→∞) [Cn+1(n+r+1)(n+r)-Cn(n+r+1)]xn+r

    Inside the brackets = 0 so the recurrence relation is

    Cn=Cn+1(n+r) , n=0,1,2,3...

    For r=1, Cn=Cn+1(n+1)

    Co=C1, n=0
    C1=C2(2)=Co/2, n=1
    C2=C3(3)=Co/2*3, n=2
    C3=C4(4)=Co/2*3*4, n=3

    I conclude y1=Co(1+x+x2/2!+x3/3!...)
    which is the series for ex, though our professor wants this in series form.

    For r=0, Cn=Cn+1(n)

    Co=0, n=0
    C1=C2, n=1
    C2=C3(2)=C1/2, n=2
    C3=C4(3)=C1/2*3, n=3

    I'm not to sure how put this into summing terms, the zero is throwing me off.
    I'd like to know if I'm on the right track with this. I feel like I did everything as I was supposed to, but something is giving me gut feeling that some portion is erroneous.

    Thanks for the help, if you choose to lend it to this tedious problem lol :zzz:
     
  2. jcsd
  3. May 13, 2012 #2
    Bumpity. Does anyone even have perhaps a hint that something is wrong?
     
  4. May 14, 2012 #3

    vela

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    Your conclusion is wrong. You can see this if you plug ex into the original differential equation. It's not a solution. Remember you're working on the case where r=1. What's the power of x in the lowest-order term?

    C0 can't be equal to 0. By definition, it's the coefficient of the lowest-order non-vanishing term. Because the two values of r differ by an integer, this method won't give you a second independent solution. You'll have to find the second solution another way.
     
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