Results 1 to 7 of 7

Math Help - ODE solved using Euler's equation

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    18

    ODE solved using Euler's equation

    This problem is getting messy fast and I'm not sure if I am doing it correctly. Any advice will be very appreciated.

    y is a function of x

    x^2y'' + 7xy'+13y=0; y(-1) = 1; y'(-1)=3

    using x = e^t, t = ln(x) and y(x) = Y(t) this becomes

    Y'' +6Y'+13Y = 0

    The characteristic equation has roots -3\pm{i2}

    so Y = c_1e^{-3t}e^{i2t} + c_2e^{-3t}e^{-i2t}

    back substitution yields:

    y = (c_1+c_2)x^{-3}cos(2ln(x)) +j(c_1-c_2)x^{-3}sin(2ln(x))

    This is about where I ran out of steam. I had started to try and choose my constants to give me two separate equations which should be linearly independent which gives me the following:

    y = k_{1}x^{-3}cos(2ln(x))+k_{2}x^{-3}sin(2ln(x))

    but plugging in my initial conditions gets so messy taking the log of negative numbers giving me imaginary arguments to sinusoids which gives hyperbolic functions.

    None of my other problems have been this tough and I'm guessing I'm doing something wrong. Thanks for any help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Hi

    This is normal : when using x = e^t, you assume that x > 0 which is not the case since obviously y(-1) and y'(-1) are defined

    Use x = -e^t can help you
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2009
    Posts
    18
    Excellent, I'll try that out and see how it works. Thanks


    edit: solved it
    Last edited by stevedave; January 20th 2009 at 10:49 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    I can tell you that it works pretty well !
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Quote Originally Posted by stevedave View Post
    edit: solved it
    Could you post your result ?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jan 2009
    Posts
    18
    Quote Originally Posted by running-gag View Post
    Could you post your result ?

    y=-x^{-3}cos(2ln(-x))
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Quote Originally Posted by stevedave View Post
    y=-x^{-3}cos(2ln(-x))
    I find the same as you
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Euler's equation
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: August 30th 2010, 05:03 AM
  2. [SOLVED] Euler Method
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: July 14th 2009, 09:46 PM
  3. [SOLVED] Two questions about Euler
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 18th 2009, 06:47 PM
  4. [SOLVED] Number Theory:Euler phi function proofs
    Posted in the Number Theory Forum
    Replies: 8
    Last Post: February 20th 2009, 01:39 AM
  5. [SOLVED] Euler Expression
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 16th 2009, 01:38 PM

Search Tags


/mathhelpforum @mathhelpforum