1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Using Undetermined Coefficients to solve an equation for a particular solution?

  1. Oct 18, 2014 #1
    1. The problem statement, all variables and given/known data
    [itex] y'' + 9y = 3sin(3x) + 3 + e^{3x}[/itex]

    2. Relevant equations

    3. The attempt at a solution
    This is my first post here so let me know if I've done anything wrong, I've been looking at questions here for a long time though ^^.

    So the problem asks me to solve for one particular solution by using undetermined coefficients, I begin by solving for the homogeneous solution:
    y'' + 9y = 0
    The characteristic polynomial becomes:
    [itex] r^2 + 9 = 0 [/itex]
    Therefore, I get the two roots [itex] {0 \pm 3i} [/itex]

    Solving for the homogeneous solution, which I'll call [itex] y_h [/itex] I get:
    [itex] y_h = cos(3x) + sin(3x) [/itex]

    Now, my professor hasn't gone through undetermined coefficients in his class yet but put it on the assignment (weird right?), so this is the part I might have screwed up on due to lack of knowledge.

    Based on what I've found on the internet, first I predict a form of the particular solution, which I will denote [itex] y_p [/itex]
    [itex] y_p = A(cos(3x)) + B(sin(3x)) + C + D(e^{3x}) [/itex]
    Afterwards, I plug this particular solution back into my original equation in order to solve for the coefficients A, B, C, and D.

    However, taking the second derivative of my particular equation, I get:
    [itex] y'' = -9A cos(3x) - 9B sin(3x) + 9D e^{3x} [/itex]
    That equation is added onto 9y, which is:
    [itex] 9y = 9Acos(3x) + 9B sin(3x) + 9C + 9D e^{3x} [/itex]

    I'm left with the final equation of:
    [itex] 9C + 18D = 3sin(3x) + 3 + e^{3x} [/itex]

    Now what I'm confused about is the fact that A and B cancel out in this case, and I've been looking at internet explanations of undetermined coefficients but I can't find an answer to this question. Can anyone help me?
  2. jcsd
  3. Oct 18, 2014 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    ##\cos(3x)## and ##\sin(3x)## satisfy the homogeneous equation so they can't contribute to the non-homogeneous. That's why they are dropping out.

    Use ##Ax\cos(3x) + Bx\sin(3x)## in your trial solution for ##y_p##.
  4. Oct 18, 2014 #3
    because your yh is Acos3x+Bsin3x, your yp should be x(Acos3x+Bsin3x).
  5. Oct 18, 2014 #4
  6. Oct 19, 2014 #5
    Ahh, I see, it was just my ignorance about this method. Thanks guys! I hope to be posting here more in the future.
  7. Oct 19, 2014 #6
    Paul's math notes has a good explanation of this method of multiplying throughout by a variable fixes problems sometimes.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted