Results 1 to 2 of 2

Math Help - differential equation using method of undetermined coefficients

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    158

    differential equation using method of undetermined coefficients

    y'' + y = e^x + x^3, y(0) = 2, y'(0)=0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by twilightstr View Post
    y'' + y = e^x + x^3, y(0) = 2, y'(0)=0
    There are two parts to this problem.

    First we need to find the complimentery solution(the solution to the homogenious solution)

    y''+y=0 has the auxillary equation

    m^2+1=0 \iff m=\pm i so

    y_c=c_1\cos(t)+c_2\sin(t)

    Now we need to find the particular solution

    y_p=Ae^{x}+B+Cx+Dx^2+Ex^3

    y_p'=Ae^{x}+C+2Dx+3Ex^2

    y_p''=Ae^{x}+2D+6Ex

    Plugging these in we get

    Ae^{x}+B+Cx+Dx^2+Ex^3+Ae^{x}+2D+6Ex=e^{x}+x^3

    2Ae^{x} +Ex^3+Dx^2 +(6E+C)x+(2D+B)=e^{x}+x^3

    From this we see that both D=0,B=0

    2A=1 \iff A=\frac{1}{2}

    E=1 and

    6E+C=0 \iff 6+C=0 \iff C=-6

    y_p=\frac{1}{2}e^{x}+x^3-6x

    So the solution by superposition

    y=y_c+y_p=c_1\cos(t)+c_2\sin(t)+\frac{1}{2}e^{x}+x  ^3-6x

    From here just plug in you I.C and you are done
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Method of undetermined coefficients
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: November 29th 2011, 03:59 PM
  2. solve the differential equation by undetermined coefficients
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: March 15th 2011, 11:41 AM
  3. Method of Undetermined Coefficients
    Posted in the Differential Equations Forum
    Replies: 8
    Last Post: May 5th 2010, 04:13 AM
  4. non-homogeneous equation, method of undetermined coefficients
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: March 4th 2010, 11:53 AM
  5. Replies: 8
    Last Post: March 14th 2009, 06:29 PM

Search Tags


/mathhelpforum @mathhelpforum