Results 1 to 7 of 7
Like Tree5Thanks
  • 2 Post By MarkFL
  • 1 Post By MarkFL
  • 2 Post By MarkFL

Math Help - Finding a particular solution using the Method of Undetermined Coefficients

  1. #1
    Junior Member
    Joined
    Sep 2010
    Posts
    64

    Finding a particular solution using the Method of Undetermined Coefficients

    Hi all,

    I have been trying to solve this problem for hours and have looked at many examples, but I still can't solve it. It is a past exam question, so I don't know if there was an error or if I am just doing it wrong.

    The problem says:

    "Consider the ordinary differential equation

    y''-2y'+y=te^t+4

    Using the Method of Undetermined Coefficients, find a particular solution y_P(t) of the inhomogeneous problem."

    So I have tried to solve it using y_P=Ate^t+Be^t+C=e^t(At+B)+C, but I can't get anything because the only result I get is C=te^t+4, and no values for A or B. And y_P=te^t+4 would give y''-2y'+y=4 instead of te^t+4.

    I would really appreciate it if someone could give me a hand. Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Finding a particular solution using the Method of Undetermined Coefficients

    You are not using the correct form for the particular solution, because the homogeneous solution is

    y_h(t)=c_1e^t+c_2te^t

    Hence, you need to try a particular solution of the form:

    y_p(t)=t^2(At+B)e^t+C

    Do you understand why?
    Thanks from topsquark and juanma101285
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2010
    Posts
    64

    Re: Finding a particular solution using the Method of Undetermined Coefficients

    Thanks a lot! I actually don't understand why though... Where does t^2 come from?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Finding a particular solution using the Method of Undetermined Coefficients

    No term in the particular solution can be a solution to the corresponding homogeneous solution, and so that's why the factor t^2 is necessary. Have you studied the annihilator method yet?
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Sep 2010
    Posts
    64

    Re: Finding a particular solution using the Method of Undetermined Coefficients

    No, I haven't... But I've just googled it and found some good files. Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Finding a particular solution using the Method of Undetermined Coefficients

    If we observe that:

    \frac{d}{dt}\left(te^t+4 \right)-\left(te^t+4 \right)=e^t-4

    \frac{d}{dt}\left(e^t-4 \right)-\left(e^t-4 \right)=4

    \frac{d}{dt}\left(4)=0

    we may then state that the differential operator:

    A\equiv D(D-1)^2

    annihilates te^t+4

    Now, applying this to the given ODE, we have:

    D(D-1)^4[y]=0

    Hence, the general solution must be of the form:

    y(t)=c_1+c_2e^t+c_3te^t+c_4t^2e^t+c_5t^3e^t

    But, we see that the homogeneous solution to the original ODE is:

    y_h(t)=c_2e^t+c_3te^t

    and we know the general solution is the superposition of the homogeneous and particular solutions:

    y(t)=y_h(t)+y_p(t)

    Hence, we must conclude that the particular solution is of the form:

    y_p(t)=c_1+c_4t^2e^t+c_5t^3e^t=t^2\left(c_4+c_5t \right)e^t+c_1
    Last edited by MarkFL; July 25th 2013 at 10:18 AM.
    Thanks from juanma101285 and topsquark
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Sep 2010
    Posts
    64

    Re: Finding a particular solution using the Method of Undetermined Coefficients

    Sorry, I was out of town...

    We had not been taught about that, but thanks so much! That makes everything a lot clearer now.
    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, 04:59 PM
  2. Method of Undetermined Coefficients
    Posted in the Differential Equations Forum
    Replies: 8
    Last Post: May 5th 2010, 05:13 AM
  3. Method of Undetermined coefficients
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: October 23rd 2009, 07:05 PM
  4. Method of Undetermined Coefficients
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: March 10th 2009, 08:34 PM
  5. Method of Undetermined Coefficients
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 15th 2008, 03:48 PM

Search Tags


/mathhelpforum @mathhelpforum