Results 1 to 2 of 2

Math Help - Finding the PS (of a DE)

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    103

    Finding the PS (of a DE)

    Use a suitable trial function to find the particular solution of

    \frac{dy}{dx}+6y=3sin(4x)

    y=Acos(4x)+Bsin(4x)

    \frac{dy}{dx}=-4Asin(2x)+4Bcos(2x)

    From the equation, -4Asin(2x)+4Bcos(2x)+6[Acos(2x)+Bsin(2x)]=3sin(4x)

    Now I don't understand the last bit where you have to pick the two parts which correspond to 0 and 3. Could someone explain it for me?
    Last edited by Haris; February 11th 2009 at 12:28 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Feb 2009
    Posts
    4
    For an equation of the form:
    \mathsf{\frac{dy}{dx}+P(x)\cdot y = Q(x)} (in this case \mathsf{P(x)=6} and \mathsf{Q(x)=5\sin{2x}})

    \mathsf{y=\frac{\int Q(x)\cdot e^{\int P(x)dx}dx}{e^{\int P(x)dx}}}

    For this case:

    \mathsf{\int P(x)dx=6x} and hence \mathsf{e^{\int P(x)dx}=e^{6x}}

    So to the guts of the problem (remembering the constant of integration at this stage):

    \mathsf{y=\frac{\int 5\sin{2x}\cdot e^{6x}dx}{e^{6x}}=\frac{e^{6x}\cdot \frac{3}{4}\sin{2x}-e^{6x}\cdot \frac{1}{4}\cos{2x}+C}{e^{6x}}=\frac{3}{4}\sin{2x}-\frac{1}{4}\cos{2x}+Ce^{-6x}}

    \mathsf{\frac{dy}{dx}=\frac{3}{2}\cos{2x}+\frac{1}  {2}\sin{2x}-6Ce^{-6x}}

    Checking:

    \mathsf{\frac{dy}{dx}+6y=5\sin{2x}}

    \mathsf{\frac{3}{2}\cos{2x}+\frac{1}{2}\sin{2x}-6Ce^{-6x}+6(\frac{3}{4}\sin{2x}-\frac{1}{4}\cos{2x}+Ce^{-6x})=5\sin{2x}}

    To determine C you'll need a known point
    Last edited by danothy; February 10th 2009 at 04:05 PM. Reason: Typo noticed in original post
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: December 8th 2011, 10:27 AM
  2. Replies: 1
    Last Post: July 3rd 2010, 10:40 PM
  3. Finding a limit. Finding Maclaurin series.
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 18th 2010, 10:04 PM
  4. Finding the radius, solving, and finding theta?
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: June 13th 2009, 02:37 PM
  5. Replies: 1
    Last Post: April 9th 2009, 09:02 AM

Search Tags


/mathhelpforum @mathhelpforum