Results 1 to 4 of 4

Math Help - probably very simple differential equation :D

  1. #1
    Junior Member
    Joined
    Sep 2010
    Posts
    47

    probably very simple differential equation :D

    hello,

    i have problem solving this one

    y'' - 2y' = e^{-x} (2\cos {x} + 3 \sin {x} )

    okay i don't have any problem with homogeneous part

    r^2-2r=0

    r(r-2)=0

    r_1=0 \; ; \; r_2=2

     y_h = C_1 e^{0x} + C_2 e^{2x}

     y_h = C_1 + C_2 e^{2x}

    i have problem with particular part now... (never encountered with similar till now) and really don't know where to go from here ...

    i know that if instead of what i have here is :

     y'' -2y' = \cos {x} + e^x

    particular part would be

     y_p = A\cos{x} + B \sin {x} + Ce^x

    but with that up there i'm just stuck ...

    any help is very much appreciated
    Last edited by sedam7; September 12th 2010 at 03:11 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member yeKciM's Avatar
    Joined
    Jul 2010
    Posts
    456
    Quote Originally Posted by sedam7 View Post
    hello,

    i have problem solving this one

    y'' - 2y' = e^{-x} (2\cos {x} + 3 \sin {x} )

    okay i don't have any problem with homogeneous part

    r^2-2r=0

    r(r-2)=0

    r_1=0 \; ; \; r_2=2

     y_h = C_1 e^{0x} + C_2 e^{2x}

     y_h = C_1 + C_2 e^{2x}

    i have problem with particular part now... (never encountered with similar till now) and really don't know where to go from here ...

    i know that if instead of what i have here is :

     y'' -2y' = \cos {x} + e^x

    particular part would be

     y_p = A\cos{x} + B \sin {x} + Ce^x

    but with that up there i'm just stuck ...

    any help is very much appreciated

    for solving differential equation formed like this

     a_0(x)y^{(n)} + a_1(x) y^{(n-1)}+ ... + a_n(x)y = e^{\alpha x } (a \cos{\beta}x + b \sin {\beta} x )

    than you should do next ...

    as you know  y = y_h +y_p

    so you should solve for  y_h first (as you did)

    now if solutions of homogeneous part are not  r = \alpha \pm i \beta (means that you don't have them in your  f(x) solution of particular part is searched as :

     y_p(x) = e^{\alpha x } (A \cos{\beta}x + B \sin {\beta} x )

    but if you have  r = \alpha \pm i \beta and are the same as f(x) than it's (because of multiplicity)

     y_p(x) =x^m e^{\alpha x } (A \cos{\beta}x + B \sin {\beta} x )
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2010
    Posts
    47
    meaning that i just look if "r" appear in homogeneous solutions.I don't look at separate  \alpha and \beta ?

    lol that would mean it's the same i'm just to dumb to see it thank you very much !
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member yeKciM's Avatar
    Joined
    Jul 2010
    Posts
    456
    Quote Originally Posted by sedam7 View Post
    meaning that i just look if "r" appear in homogeneous solutions.I don't look at separate  \alpha and \beta ?

    lol that would mean it's the same i'm just to dumb to see it thank you very much !
    yes it's the same. As you look at the real solutions, you look at the complex solutions
    good luck
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Simple Differential Equation
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: March 21st 2011, 06:40 PM
  2. Simple differential equation
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: January 16th 2010, 08:22 AM
  3. Very simple differential equation
    Posted in the Calculus Forum
    Replies: 4
    Last Post: May 18th 2008, 05:13 PM
  4. simple differential equation
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 18th 2007, 12:13 PM
  5. A simple differential equation.
    Posted in the Calculus Forum
    Replies: 7
    Last Post: September 11th 2007, 04:33 PM

Search Tags


/mathhelpforum @mathhelpforum