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Math Help - Determining the first three nonzero terms for Taylor polynomial approximation for IVP

  1. #1
    s3a
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    Determining the first three nonzero terms for Taylor polynomial approximation for IVP

    I only know how to do this with regular equations. How do I do this with a differential equation involved?

    Any help would be greatly appreciated!
    Thanks in advance!
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  2. #2
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    Re: Determining the first three nonzero terms for Taylor polynomial approximation for

    You have not understood the definition.

    The first term requires y(0), whci is zero. It is therefore not a non-zero term.

    The second term is related to 8*sin(0) + 7*e^(0) = 7 -- This should lead you to the first non-zero term.

    The third term whould be related to y".

    You need values of the function and its derivatives. When they give you a derivative, don't be afraid of it.
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    s3a
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    Re: Determining the first three nonzero terms for Taylor polynomial approximation for

    Treating y as a constant, I get:

    y'' = 7e^x = 7e^x

    Yielding y(x) = 7x + 7x^2/2! + 7 x^3/3! but that is incorrect. I am thinking I shouldn't be treating y as constant but I don't know what else to do.
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    MHF Contributor chisigma's Avatar
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    Re: Determining the first three nonzero terms for Taylor polynomial approximation for

    Quote Originally Posted by s3a View Post
    I only know how to do this with regular equations. How do I do this with a differential equation involved?

    Any help would be greatly appreciated!
    Thanks in advance!
    Let's write again the DE...

    y^{'}= 8\ \sin y + 7\ e^{x},\ y(0)=0 (1)

    Under the assumption that the solution is analytic in x=0, we can write...

    y(x)= \sum_{n=0}^{\infty} \frac{y^{(n)}(0)}{n!}\ x^{n}= \sum_{n=0}^{\infty} a_{n}\ x^{n} (2)

    The derivatives of y in x=0 are computed as follows...

    y(0)=0 \implies a_{0}=0

    ... and from (1)...

    y^{'}(0)= 7 \implies a_{1}=7

    Now we compute y^{''} from (1)...

    y^{''}= 8\ \cos y\ y^{'} + 7\ e^{x} (3)

    ... and from (3)...

    y^{''}(0)= 8 \cdot 7 +7 = 63 \implies a_{2}= \frac{63}{2}

    Now You can proceed to compute succesive terms... if necessary...
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    Re: Determining the first three nonzero terms for Taylor polynomial approximation for

    Quote Originally Posted by s3a View Post
    Treating y as a constant,
    Why would you do that? y is a function of x. Please note the chain rule as demonstrated by chisigma.
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