Results 1 to 3 of 3

Math Help - differencial equation

  1. #1
    Junior Member
    Joined
    Jan 2009
    Posts
    64

    differencial equation

    Solve x^2y"-4xy'+6y=0 to obtain y=

    kindly assist me with the full answer and some explanation.

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,313
    Thanks
    1291
    That is an "equi-potential" or "Euler type" equation: each derivative is multiplied by a power of x, the degree the same as the order of the derivative.

    If you make the substitution x= ln t will convert it to a linear differential equation with constant coefficients for y as a function of t.

    For simple equation, you can try a "trial" solution of the form y= x^r for some constant r.

    If y= x^r, then y'= rx^{r-1} and y''= r(r-1)x^{r-2}. Putting those into this equation,

    x^2y''-4xy'+6y= (x^2)r(r-1)x^{r-2}- 4x(r)x^{r-1}+ 6x^r= x^r(r^2- r- 4r+ 6)= x^2(r^2- 5r+ 6)= 0.

    In order for that to be 0 for all x, we must have the "characteristic equation", r^2- 5t+ 6= (r- 3)(r- 2)= 0 so r= 3 and r= 2.
    Both x^3 and x^2 are solutions to this equation. Since it is a linear homogeneous equation, the general solution is y(x)= Cx^2+ Dx^3 where C and D can be any constants.
    Last edited by mr fantastic; January 10th 2009 at 04:39 AM. Reason: Fixed a small bit of latex - getting better, by George lol!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jan 2009
    Posts
    64

    Thanks

    Thanks so much.

    I'll try and understand it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. differencial equation
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 8th 2008, 11:03 AM
  2. differencial equation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 5th 2008, 10:31 AM
  3. differencial equation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 28th 2008, 04:52 AM
  4. differencial equation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 20th 2007, 04:14 PM
  5. differencial equation
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 31st 2007, 08:37 AM

Search Tags


/mathhelpforum @mathhelpforum