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Math Help - just a quick particular integral question..

  1. #1
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    just a quick particular integral question..

    if i have a problem...

    <br />
y'' + 2y' + 4 = x^2 e^{ - 2x} <br />

    would the substitution be..

    <br />
y(x) = e^{ - 2x} v<br />


    I know it would be for = xe^-2x..
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  2. #2
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    Quote Originally Posted by dankelly07 View Post
    if i have a problem...

    <br />
y'' + 2y' + 4 = x^2 e^{ - 2x} <br />

    would the substitution be..

    <br />
y(x) = e^{ - 2x} v<br />


    I know it would be for = xe^-2x..
    Quick question is it supposed to be

    <br />
y'' + 2y' + 4{\color{red}y} = x^2 e^{ - 2x} <br />

    Or what you have typed above
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  3. #3
    MHF Contributor

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    Quote Originally Posted by dankelly07 View Post
    if i have a problem...

    <br />
y'' + 2y' + 4 = x^2 e^{ - 2x} <br />

    would the substitution be..

    <br />
y(x) = e^{ - 2x} v<br />


    I know it would be for = xe^-2x..
    It's not clear what you are asking. You seem to be conflating "variation of parameters" and "undetermined coefficients".

    Assuming you mean y"+ 2y'+ 4y= x^2e^{-2x}, which has e^{-2x} and xe^{-2x} as independent solutions to the associated homogenous equation, in order to use "variation of parameters" you would have to use y(x)= u(x)e^{-x}+ v(x)x^2e^{-x}. If you are referring to "undetermined coefficients", you would try y(x)= (Ax^2+ bx+ C)x^2e^{-x}= (Ax^4+ Bx^3+ Cx^2)e^{-x}.
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