Results 1 to 2 of 2

Math Help - Find the second solution of a differential equation

  1. #1
    Newbie
    Joined
    May 2008
    Posts
    22

    Find the second solution of a differential equation

    Need help...Given a differential equation and its first solution y1 find the second solution y2 using reduction of order.

    x^2 y'' - 3xy' +5y =0; y1 = x^2 cos (ln x)

    thank you so much!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,184
    Thanks
    403
    Awards
    1
    Quote Originally Posted by kithy View Post
    Need help...Given a differential equation and its first solution y1 find the second solution y2 using reduction of order.

    x^2 y'' - 3xy' +5y =0; y1 = x^2 cos (ln x)

    thank you so much!
    Let y_2(x) = v(x) \cdot y_1(x)

    Then
    y_2' = v'y_1 + vy_1'

    y_2'' = v''y_1 + 2v'y_1 + vy_1''

    x^2(v''y_1 + 2v'y_1 + vy_1'') -3x(v'y_1 + vy_1') + 5(vy_1) = 0

    Now look at the term proportional to v(x):
    y_1'' - 3xy' + 5y_1

    But this is just 0 according to the differential equation. So we get
    x^2y_1v'' + (2x^2y' - 3xy)v' = 0

    and you can take it from here.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 8
    Last Post: April 20th 2011, 12:03 AM
  2. Find the general solution of the differential equation specified..
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: March 14th 2011, 03:39 AM
  3. Find the particular solution of the differential equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: November 15th 2010, 02:29 AM
  4. Find a differential equation given solution
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: September 21st 2010, 12:15 AM
  5. To find a solution to this Differential Equation?
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: February 17th 2010, 02:43 PM

Search Tags


/mathhelpforum @mathhelpforum