Results 1 to 3 of 3

Math Help - Solving differential equation

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    297

    Solving differential equation

    Solve x^2y''+xy'+y=\log x \ \ \ \ x>0 by letting x=e^t

    How do I find y' and y''?

    \frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}

    \frac{dy}{dx}=\frac{dy}{dt}\cdot e^{-t}

    Not too sure what to do next?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    10,969
    Thanks
    1011
    Your calculation of \frac{dy}{dx} is correct.

    So \frac{d^2y}{dx^2} = \frac{d}{dx}\left(e^{-t}\,\frac{dy}{dt}\right)

     = e^{-t}\,\frac{d}{dx}\left(\frac{dy}{dt}\right) + \frac{d}{dx}(e^{-t})\,\frac{dy}{dt}

     = e^{-t}\,\frac{d}{dt}\left(\frac{dy}{dt}\right)\,\frac{  dt}{dx} + \frac{d}{dt}(e^{-t})\,\frac{dt}{dx}\,\frac{dy}{dt}

     = e^{-2t}\,\frac{d^2y}{dt^2} - e^{-2t}\,\frac{dy}{dt}.


    Now substitute everything into your DE.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Feb 2008
    Posts
    297
    Quote Originally Posted by Prove It View Post
    Your calculation of \frac{dy}{dx} is correct.

    So \frac{d^2y}{dx^2} = \frac{d}{dx}\left(e^{-t}\,\frac{dy}{dt}\right)

     = e^{-t}\,\frac{d}{dx}\left(\frac{dy}{dt}\right) + \frac{d}{dx}(e^{-t})\,\frac{dy}{dt}

     = e^{-t}\,\frac{d}{dt}\left(\frac{dy}{dt}\right)\,\frac{  dt}{dx} + \frac{d}{dt}(e^{-t})\,\frac{dt}{dx}\,\frac{dy}{dt}

     = e^{-2t}\,\frac{d^2y}{dt^2} - e^{-2t}\,\frac{dy}{dt}.


    Now substitute everything into your DE.
    Thanks, that makes a lot of sense now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solving Differential Equation
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: February 21st 2011, 06:20 PM
  2. solving a differential equation.
    Posted in the Differential Equations Forum
    Replies: 7
    Last Post: May 13th 2010, 01:52 PM
  3. Help for solving the differential equation!!
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: September 25th 2009, 11:59 AM
  4. Solving differential equation
    Posted in the Calculus Forum
    Replies: 5
    Last Post: March 26th 2009, 02:36 AM
  5. Solving this differential equation..
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 12th 2007, 07:09 PM

Search Tags


/mathhelpforum @mathhelpforum