Results 1 to 8 of 8

Math Help - [SOLVED] How do you solve y'' + (y')/x = 0

  1. #1
    Junior Member
    Joined
    Apr 2009
    Posts
    70

    [SOLVED] How do you solve y'' + (y')/x = 0

    I have the following homogenous 2nd order diff. eqn that has me stumped.

    y'' + y'/x = 0

    I don't know how to get the general soln. Any help would be greatly appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    Quote Originally Posted by spearfish View Post
    I have the following homogenous 2nd order diff. eqn that has me stumped.

    y'' + y'/x = 0

    I don't know how to get the general soln. Any help would be greatly appreciated.
    Let y' = u. The ODE will separate. Then integrate again.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2009
    Posts
    70
    Thanks Danny,

    Ok, so if I let y' = u, then y'' = u', so I get u' + u/x = 0. Then I do u' = -u/x, Integrate to get U = -U^2/(2x).

    Where do I go from here? I don't even know if what I did is valid? I am just really confused on this one.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    Quote Originally Posted by spearfish View Post
    Thanks Danny,

    Ok, so if I let y' = u, then y'' = u', so I get u' + u/x = 0. Then I do u' = -u/x, Integrate to get U = -U^2/(2x).

    Where do I go from here? I don't even know if what I did is valid? I am just really confused on this one.
    No. Separate \frac{du}{u} = - \frac{dx}{x}.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Apr 2009
    Posts
    70
    ahhh, now it's coming back to me. I ll work on in it some more and post back later. Thanks
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,407
    Thanks
    1294
    Alternatively, you can use the Integrating Factor.

    \frac{du}{dx} + \frac{1}{x}u = 0


    Integrating Factor: e^{\int{\frac{1}{x}\,dx}} = e^{\ln{x}} = x.

    Multiply both sides by the integrating factor:

    x\frac{du}{dx} + u = 0

    \frac{d}{dx}(ux) = 0

    ux = \int{0\,dx}

    ux = C

    u = \frac{C}{x}.


    Now since u = \frac{dy}{dx}

    \frac{dy}{dx} = \frac{C}{x}

    y = \int{\frac{C}{x}\,dx}

    y = C\ln{|x|} + D.
    Last edited by Prove It; December 14th 2009 at 02:42 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    As alternative You can use the ever useful change on variable u=\ln x illustrated in...

    http://www.mathhelpforum.com/math-he...-question.html

    We have...

    \frac{dy}{dx} = \frac{1}{x}\cdot \frac{dy}{du}

    \frac{d^{2}y}{dx^{2}}=\frac{1}{x^{2}}\cdot (\frac{d^{2}y}{du^{2}} - \frac{dy}{du}) (1)

    ... so that ther DE...

    \frac{d^{2}y}{dx^{2}} + \frac{1}{x}\cdot \frac{dy}{dx}=0 (2)

    ...becomes...

    \frac{d^{2}y}{du^{2}} =0 (3)

    The integration of (3) is immediate and we obtain...

    y=c_{1}\cdot u + c_{2} = c_{1}\cdot \ln x + c_{2} (4)



    Merry Christmas from Italy

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Apr 2009
    Posts
    70
    Thanks guys.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Solve for a DE
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: February 22nd 2010, 01:26 PM
  2. [SOLVED] Please help solve for d
    Posted in the Algebra Forum
    Replies: 12
    Last Post: September 14th 2009, 10:52 PM
  3. [SOLVED] Need to solve
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: July 17th 2009, 05:11 AM
  4. [SOLVED] Solve: ln x + ln(x+2) = ln(x + 6)
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: February 5th 2009, 09:41 AM
  5. [SOLVED] Please solve this...
    Posted in the Calculus Forum
    Replies: 6
    Last Post: March 1st 2008, 08:57 AM

Search Tags


/mathhelpforum @mathhelpforum