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Math Help - Maclaurin Series for a Differential Equation

  1. #1
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    Maclaurin Series for a Differential Equation

    Consider the differential equation:
    \frac{dy}{dx} = y - x + 1
    The initial condition is y(0) = 1. Construct the Maclaurin series for y(x).

    I know I'm supposed to differentiate it, but I don't remember how to use the initial condition to start me off...
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  2. #2
    MHF Contributor chisigma's Avatar
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    The McLaurin series for y is...

    y= \sum_{n=0}^{\infty} a_{n}\cdot x^{n} (1)

    ... so that...

    y^{'}=  \sum_{n=0}^{\infty} n\cdot a_{n-1}\cdot x^{n-1} (2)

    The 'initial condition' y(0)=1 extablishes that a_{0}=1. If now write the DE in terms of (1) and (2) we obtain...

    \sum_{n=0}^{\infty} \{(n+1)\cdot a_{n+1} -a_{n}\}\cdot x^{n}= 1-x (3)

    .. so that is...

    a_{1}=2

    a_{2}=\frac{1}{2}

    a_{n+1}= \frac{a_{n}}{n+1}, n>2 (4)

    Kind regards
    Last edited by chisigma; February 11th 2010 at 09:00 AM. Reason: some minor errors...
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  3. #3
    MHF Contributor Calculus26's Avatar
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    see attachment
    Attached Thumbnails Attached Thumbnails Maclaurin Series for a Differential Equation-desol.jpg  
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