1. Not finding help here? Sign up for a free 30min 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!

Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0

  1. Aug 8, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0.

    2. Relevant equations
    None.

    3. The attempt at a solution
    I'll post my whole work.
     
  2. jcsd
  3. Aug 8, 2016 #2
    This is my work, I have more to post.
     

    Attached Files:

    • MM.jpg
      MM.jpg
      File size:
      43.2 KB
      Views:
      47
  4. Aug 8, 2016 #3
    This work comes first, the above one comes after this one.
     

    Attached Files:

    • MMM.jpg
      MMM.jpg
      File size:
      41.9 KB
      Views:
      50
  5. Aug 8, 2016 #4
    But I don't know how to get to the book's answer.
     

    Attached Files:

    • de3.jpg
      de3.jpg
      File size:
      26 KB
      Views:
      49
  6. Aug 9, 2016 #5

    pasmith

    User Avatar
    Homework Helper

    You have the correct recurrence relation [tex]a_{2n} = - \frac{2n+1}{n+1}a_n.[/tex] You just haven't tried to solve it.

    Recurrences of the form [tex]
    a_{n+1} = f(n)a_n[/tex] have the solution [tex]
    a_n = a_0\prod_{k=0}^{n-1} f(k)[/tex] where by convention [tex]
    \prod_{k=0}^{-1} f(k) = 1.[/tex]
    Recurrences of the form [tex]
    a_{n+2} = f(n)a_n[/tex] can be turned into the above form by treating even and odd terms separately: First set [itex]n = 2m[/itex] and [itex]b_m = a_{2m}[/itex] to obtain [tex]
    b_{m+1} = f(2m)b_m[/tex] and then set [itex]n = 2m+1[/itex] and [itex]c_m = a_{2m+1}[/itex] to obtain [tex]
    c_{m+1} = f(2m+1)c_m.[/tex]
     
  7. Aug 10, 2016 #6
    But that's not the answer in the book. How do I get the answer in the book?
     
  8. Aug 11, 2016 #7

    pasmith

    User Avatar
    Homework Helper

    Are you saying that [tex]
    y(x) = \sum_{n=0}^\infty a_nx^n = \sum_{m=0}^\infty b_mx^{2m} + \sum_{m=0}^\infty c_mx^{2m+1}[/tex] with [itex]b_m[/itex] and [itex]c_m[/itex] obtained as I have suggested is not the answer in the book?

    What do you get for [itex]b_m[/itex] and [itex]c_m[/itex]?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted