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Finding particular solution to differential equations

  1. Oct 16, 2012 #1
    There are two questions that I am trying to solve on web assignment. The goal is to find a general form of a particular solution to each ODE. The question asks me to represent all constants in the solution using "P,Q,R,S,T..etc.", in that order.

    1. y'''-9y''+14y'=x2
    2. y''-9y'+14y=x2e4x

    For the first one I wrote:
    yp=Px3+Qx2+Rx

    Second one:
    yp=(Px2+Qx+R)e4x

    Neither of the answers are correct, according to the computer. Where did I go wrong?

    Thanks.
     
  2. jcsd
  3. Oct 16, 2012 #2

    Zondrina

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    The answers are not correct because you're not solving the equations correctly.... First do you understand that your equations are non-homogenous?
     
  4. Oct 16, 2012 #3
    Yes, of course, that's why I am asked to find a particular solution. Where did I go wrong? please let me know!
     
  5. Oct 16, 2012 #4

    Zondrina

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    Okay, so take for example your second question. Notice that your derivatives and original function have to add up to x2e4x?

    So your general solution must have the form : y = A(x2e4x) where A is some constant number.

    Now take the required derivatives and see if they satisfy your equation. ( You'll get a bunch of constants which you have to add together and solve for A for the particular solution you want ).
     
  6. Oct 16, 2012 #5
    I don't quite understand why your guess for the particular solution has the form Ax2e4x as opposed to (Ax2+Bx+C)e4x. Shouldn't you use the general form of the second degree polynomial?
     
  7. Oct 16, 2012 #6

    Zondrina

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    A non-homogenous equation with constant coefficients has the general solution y = Ag(t) where A is a constant.
     
  8. Oct 16, 2012 #7
    Then suppose for y''-9y'+14y my yp=Ax2e4x

    Then

    y'=2Axe4x+4Ax2e4x
    y''=2Ae4x+8Axe4x+8Axe4x+16Ax2e4x

    When you plug this all back into the original equation, equate the coefficients to solve for A, it doesn't work. Need more than just one constant.
     
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