Results 1 to 5 of 5

Math Help - Method of variation of parameters

  1. #1
    Senior Member chella182's Avatar
    Joined
    Jan 2008
    Posts
    267

    Method of variation of parameters

    The question goes...

    Find the general solution of this differential equation, given that a solution of the corresponding homogeneous equation is y=x,
    x^2y''+x(x-2)y'-(x-2)y=x^3

    I've checked that y=x is a solution when the equation has a 0 on the RHS, but it's the whole setting y(x)=u(x)v(x) thing I'm stuck with. Obviously that's y(x)=xv(x), but from there I'm stuck. Can anyone help?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,392
    Thanks
    56
    Quote Originally Posted by chella182 View Post
    The question goes...

    Find the general solution of this differential equation, given that a solution of the corresponding homogeneous equation is y=x,
    x^2y''+x(x-2)y'-(x-2)y=x^3

    I've checked that y=x is a solution when the equation has a 0 on the RHS, but it's the whole setting y(x)=u(x)v(x) thing I'm stuck with. Obviously that's y(x)=xv(x), but from there I'm stuck. Can anyone help?
    First you want to solve the homogeneous equation (i.e. rhs = 0). This is done using what you said y = x u . Sub. this into your equation
    x^2y''+x(x-2)y'-(x-2)y=0. You'll get an equation with only u' and u''. Let u' = v which gives a first order you can solve. Then find u.

    Then use variation of parameters.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member chella182's Avatar
    Joined
    Jan 2008
    Posts
    267
    S'alright, I've solved this in the mean time. I didn't understand what I'd written in my notes, but then I remembered what it was. Thanks for the reply, though.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,454
    Thanks
    1868
    Quote Originally Posted by chella182 View Post
    The question goes...

    Find the general solution of this differential equation, given that a solution of the corresponding homogeneous equation is y=x,
    x^2y''+x(x-2)y'-(x-2)y=x^3

    I've checked that y=x is a solution when the equation has a 0 on the RHS, but it's the whole setting y(x)=u(x)v(x) thing I'm stuck with. Obviously that's y(x)=xv(x), but from there I'm stuck. Can anyone help?
    Well, if y= xv, then y'= xv'+ v and y"= xv"+ 2v'. What do you get if you put that into your equation?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member chella182's Avatar
    Joined
    Jan 2008
    Posts
    267
    Like I said, I've sorted it cheers, though.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Theoretical question on the Variation of Parameters method
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: June 26th 2011, 07:15 AM
  2. ODE Method of Variation of Parameters help
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: April 1st 2011, 05:58 AM
  3. Solving using the IF method and variation of parameters
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: November 5th 2010, 03:56 PM
  4. Replies: 1
    Last Post: October 13th 2010, 05:18 PM
  5. Solve using method of variation of parameters
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: April 29th 2009, 04:16 PM

Search Tags


/mathhelpforum @mathhelpforum