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Math Help - IVP Undetermined Coefficients

  1. #1
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    IVP Undetermined Coefficients

    Solve by using Undetermined Coefficients:

    y''+y=e^x+x^3 , y(0) = 2 , y'(0) = 0

    so I found r to equal + i which gives the first particular solution
    y(x) = C1*cosx+C2sinx

    for the first side I got yp(x) = Ae^x and both y' and y'' to equal Ae^x.

    I solved for A to get 1/2.

    can someone show me the 2nd part with the x^3? what would my yp(x) be? thanks
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  2. #2
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    Quote Originally Posted by xfyz View Post
    Solve by using Undetermined Coefficients:

    y''+y=e^x+x^3 , y(0) = 2 , y'(0) = 0

    so I found r to equal + i which gives the first particular solution
    y(x) = C1*cosx+C2sinx

    for the first side I got yp(x) = Ae^x and both y' and y'' to equal Ae^x.

    I solved for A to get 1/2.

    can someone show me the 2nd part with the x^3? what would my yp(x) be? thanks
    With the x^3 look for a solution of the form Ax^3+Bx^2+Cx+D.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    With the x^3 look for a solution of the form Ax^3+Bx^2+Cx+D.
    So I used Bx^3+Cx^2+Dx+E since I used A for the first part.

    y' = 3Bx^2+2Cx+D
    y'' = 6Bx + 2C

    (6Bx+2C) + (Bx^3+Cx^2+Dx+E) = x^3

    isn't it just x^3(B) = x^3 , B = 1?
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