Results 1 to 8 of 8

Math Help - Linear first order ODE question

  1. #1
    Senior Member
    Joined
    Apr 2010
    Posts
    487

    Linear first order ODE question

    The question:

    x^2 \frac{dy}{dx} -xy = y

    I'm not sure where to start. I tried getting it in a form where I can calculate the integrating factor, but nothing I do seems to work. Any suggestions?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Master Of Puppets
    pickslides's Avatar
    Joined
    Sep 2008
    From
    Melbourne
    Posts
    5,236
    Thanks
    28
    add xy to both sides, then divide x^2 to both sides, does it separate?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,569
    Thanks
    1428
    Quote Originally Posted by Glitch View Post
    The question:

    x^2 \frac{dy}{dx} -xy = y

    I'm not sure where to start. I tried getting it in a form where I can calculate the integrating factor, but nothing I do seems to work. Any suggestions?
    To make it linear

    \displaystyle x^2\,\frac{dy}{dx} - x\,y - y = 0

    \displaystyle x^2\,\frac{dy}{dx} - (x + 1)\,y = 0

    \displaystyle \frac{dy}{dx} - (x^{-1} + x^{-2})\,y = 0.


    Go from here.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Apr 2010
    Posts
    487
    Thanks guys, I'll give it another shot.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Apr 2010
    Posts
    487
    Quote Originally Posted by Prove It View Post
    \displaystyle \frac{dy}{dx} - (x^{-1} + x^{-2})\,y = 0.
    If I divide both sides by x^2, wouldn't I lose part of the solution? Otherwise I could divide both sides by the entire LHS and get 1 = 0. Or am I missing something?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,569
    Thanks
    1428
    You can if you assume that \displaystyle x \neq 0. It's a standard technique.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Apr 2010
    Posts
    487
    Ahh, ok. Thanks.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,569
    Thanks
    1428
    I suppose that you could always write it like this...

    \displaystyle x^2\,\frac{dy}{dx} - x\,y - y = 0

    \displaystyle x^2\left(\frac{dy}{dx} - x^{-1}y - x^{-2}y\right) = 0

    \displaystyle x^2\left[\frac{dy}{dx} - (x^{-1} + x^{-2})\,y\right] = 0 .


    You can still use the Integrating Factor Method and then solve the equation using NFL.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Simple question for linear first-order differential equation. I can't move on!
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: September 17th 2011, 05:06 PM
  2. Linear program with higher order non-linear constraints.
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: September 12th 2010, 02:36 AM
  3. Second order linear ODE
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: September 7th 2010, 10:31 PM
  4. Replies: 4
    Last Post: August 12th 2008, 04:46 AM
  5. Replies: 1
    Last Post: May 11th 2007, 03:01 AM

Search Tags


/mathhelpforum @mathhelpforum