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Differential equations power series method

  1. Apr 7, 2010 #1
    1. The problem statement, all variables and given/known data

    using the power series method (centered at t=0) y'+t^3y = 0 find the recurrence relation

    2. Relevant equations

    y= [tex]\sum a_{n}t^{n}[/tex] from n=0 to infinity
    y'= [tex]\sum na_{n}t^{n-1}[/tex] from n=1 to infinity

    3. The attempt at a solution
    I went through and solved by putting the values from b into the equation and i got down to this:
    a[tex]_{1}[/tex]+2a[tex]_{2}[/tex]t+3a[tex]_{3}[/tex]t[tex]^{2}[/tex]+[tex]\sum[na_{n}[/tex]+a[tex]_{n-4}][/tex]t[tex]^{n-1}[/tex] = 0

    the 1 2 and 3 should be subscripted on a

    I understand how to get an which i think i got right... i got a[tex]_{n}[/tex] = -a[tex]_{n-4}[/tex]/n whre n >= 4

    But I don't understand what i do with the other part.. the a[tex]_{1}[/tex]+2a[tex]_{2}[/tex]t+3a[tex]_{3}[/tex]t[tex]^{2}[/tex]
    Do i set it to 0 and solve for something? or do i set each individual component to 0. If i do ti this way and i find the power series solution by going through values of n until i find a pattern they all end up being 0 which can't be right. I'm completely lost.

    Please let me know, this is due tomorrow and I have everything done except I don't understand what to do with this one. I just need to know what to do with that part, I can do the rest.

    Thank you so much
     
  2. jcsd
  3. Apr 7, 2010 #2

    vela

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    You're right. You set the first three terms to zero, so you get [itex]a_1=a_2=a_3=0[/itex]. The only non-zero terms in your power series will therefore be for n=0, 4, 8, ....

    This differential equation is separable, so you can find the solution in closed form. Expand it as a series and see if it matches what the series solution seems to be giving you.
     
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