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Series solution to linear differential equations

  1. Nov 29, 2011 #1
    1. The problem statement, all variables and given/known data

    Given the ODE y''-ty'+y=0 where y(0)=1 and y'(0)=0
    Assume y(t)=Ʃn=0 ( a(n) t^n ) (power series centered at 0)

    find the general form of the solution ( an=f(n) )

    3. The attempt at a solution

    I used the initial conditions to determine the values a0=1 and a1=0

    Determined the recurrence relation a(n+2)= a(n) (n-1) / (n+1)(n+2)

    And found the first six non-zero terms (which was asked for in an earlier part of the question.

    a(0)=1 a(1)=0 a(2)=-1/2! a(3)=0 a(4)=-1/4! a(5)=0 a(6)=-3/6! a(7)=0 a(8)=-15/8! a(9)=0 a(10)=-105/10!

    I am having a really tough time coming up with a(n) I can identify a few patterns such as there is obviously a component 1/n! and since all the odd terms are 0 it would be 1/(n+1)! I attempted to handle the first term being positive and the rest negative using (n-1)/abs(n-1) I am assuming there is a better way to handle this however. I am very lost on coming up with a series representation of the numerator {1-1-1-3-15-105-...-...}

    Any help will be appreciated very much.
    Thanks for your time!
     
  2. jcsd
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