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Math Help - Second order differential equations!

  1. #1
    Junior Member
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    Cool Second order differential equations!

    Find y in terms of x given that
    d2y/dx2 - 4(dy/dx) + 4y = e^2x

    and that dy/dx=1 and y=0 at x=0

    I've worked out that the complementary function is (A+Bx)e^2x

    However, i don't understand why I should be using y=k(x^2)e^2x as the particular integral and not y=ke^2x

    Can anyone explain?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Erghhh View Post
    Find y in terms of x given that
    d2y/dx2 - 4(dy/dx) + 4y = e^2x

    and that dy/dx=1 and y=0 at x=0

    I've worked out that the complementary function is (A+Bx)e^2x

    However, i don't understand why I should be using y=k(x^2)e^2x as the particular integral and not y=ke^2x

    Can anyone explain?
    Since your complimentary solution contains a xe^{2x} term, then by reduction of order, your particular solution must contain a x^2e^{2x} term. Thus, I believe your particular solution must take on the form y_p=\left(A_1+A_2x+A_3x^2\right)e^{2x}.
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