Results 1 to 15 of 15
Like Tree1Thanks
  • 1 Post By BobP

Math Help - What method to solve this difference equation?

  1. #1
    Member zzizi's Avatar
    Joined
    Feb 2010
    From
    London
    Posts
    83

    What method to solve this difference equation?

    Hi

    Could someone shed some light, on how I would solve the following difference equation?

    u_{n}=\frac{5}{n}u_{n-1} + \frac{5^{n}}{n!}, u_{0}=2


    Would I use the divide and conquer method or something else?

    I look forward to your responses!
    Last edited by zzizi; December 23rd 2012 at 01:16 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,607
    Thanks
    591

    Re: What method to solve this difference equation?

    Hey zzizi.

    For these kinds of problems, you should consider getting a_n as a function of n purely and see if there is a relationship.

    Also do you have a set of identities or formulas for your class that you are doing this exercise for?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member zzizi's Avatar
    Joined
    Feb 2010
    From
    London
    Posts
    83

    Re: What method to solve this difference equation?

    Quote Originally Posted by chiro View Post
    Hey zzizi.

    For these kinds of problems, you should consider getting a_n as a function of n purely and see if there is a relationship.

    Also do you have a set of identities or formulas for your class that you are doing this exercise for?

    Hi Chiro!

    Yes I have been given a formula - I have attached it. Please have a look, I'd appreciate help in trying to apply this problem using the given formula. Thank youWhat method to solve this difference equation?-d-eqn-formula.jpg
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,607
    Thanks
    591

    Re: What method to solve this difference equation?

    Can you identify what the f's and g's should be in this formula? (It's mostly a thing of plugging in the definitions).
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member zzizi's Avatar
    Joined
    Feb 2010
    From
    London
    Posts
    83

    Re: What method to solve this difference equation?

    Quote Originally Posted by chiro View Post
    Can you identify what the f's and g's should be in this formula? (It's mostly a thing of plugging in the definitions).
    Is this correct?


    2\prod_{i=1}^{n}\frac{5}{i}+\sum_{i=1}^{n}\frac{5^  {i}}{i}.\prod_{j=i+1}^{n}\frac{5}{j}
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,607
    Thanks
    591

    Re: What method to solve this difference equation?

    What about the factorial term? [Recall that g(n) = 5^n/n!]
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member zzizi's Avatar
    Joined
    Feb 2010
    From
    London
    Posts
    83

    Re: What method to solve this difference equation?

    Oh yes!

    2\prod_{i=1}^{n}\frac{5}{i}+\sum_{i=1}^{n}\frac{5^ {i}}{i!}.\prod_{j=i+1}^{n}\frac{5}{j!}

    How about now?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,607
    Thanks
    591

    Re: What method to solve this difference equation?

    Recall that f(n) = 5/n [Hint: Look at the last multiplication term, you've confused f(n) and g(n) using the definition]
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member zzizi's Avatar
    Joined
    Feb 2010
    From
    London
    Posts
    83

    Re: What method to solve this difference equation?

    Quote Originally Posted by chiro View Post
    Recall that f(n) = 5/n [Hint: Look at the last multiplication term, you've confused f(n) and g(n) using the definition]
    2\prod_{i=1}^{n}\frac{5}{i}+\sum_{i=1}^{n}\frac{5^ {i}}{i!}.\prod_{j=i+1}^{n}\frac{5}{j}

    Thank you Chiro - I appreciate your patience and help. How about now?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,607
    Thanks
    591

    Re: What method to solve this difference equation?

    That looks pretty good.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member zzizi's Avatar
    Joined
    Feb 2010
    From
    London
    Posts
    83

    Re: What method to solve this difference equation?

    Would this next step be correct?

    2.\frac{5^{n}}{n!}+\sum_{i=1}^{n}\frac{5^{i}}{i!}.  \frac{5^{n}i!}{n!5^{i}}
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Member zzizi's Avatar
    Joined
    Feb 2010
    From
    London
    Posts
    83

    Re: What method to solve this difference equation?

    Can anyone help me with this next step? or possibly offer a link to site that offers a tutorial on this subject. Is this called the telescoping method? Many thanks in advance
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Super Member
    Joined
    Jun 2009
    Posts
    652
    Thanks
    128

    Re: What method to solve this difference equation?

    First solve the homogeneous equation

    w_{n}=\frac{5}{n}w_{n-1}.

    Work it out from

    w_{1}= \frac{5}{1}w_{0},

    w_{2}=\frac{5}{2}w_{1}=\frac{5^{2}}{2!}w_{0},

    w_{3}=\frac{5}{3}w_{2}=\frac{5^{3}}{3!}w_{0},

    etc.

    Having done that, assume that the solution to the original equation is u_{n}=v_{n}w_{n}.

    Substitute that in and show that v_{n}=1+\frac{n}{2}.

    That gets you the solution u_{n}=(2+n)\frac{5^{n}}{n!}.
    Thanks from zzizi
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Member zzizi's Avatar
    Joined
    Feb 2010
    From
    London
    Posts
    83

    Re: What method to solve this difference equation?

    Thank you so so much! I wish you were my lecturer!!
    Last edited by zzizi; December 25th 2012 at 02:59 PM.
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Member zzizi's Avatar
    Joined
    Feb 2010
    From
    London
    Posts
    83

    Re: What method to solve this difference equation?

    .
    Last edited by zzizi; December 25th 2012 at 09:23 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Difference equation - struggling to solve the equation below
    Posted in the Advanced Applied Math Forum
    Replies: 3
    Last Post: June 10th 2012, 02:49 PM
  2. Solve Linear Difference equation
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: August 6th 2011, 01:39 AM
  3. Z-transform to solve difference equation
    Posted in the Differential Equations Forum
    Replies: 10
    Last Post: June 10th 2010, 06:44 AM
  4. Replies: 0
    Last Post: April 7th 2010, 09:41 AM
  5. Solve The Difference Equation
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: April 30th 2009, 12:52 AM

Search Tags


/mathhelpforum @mathhelpforum