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Thread: ODEs with a particular solution

  1. #1
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    ODEs with a particular solution

    You are given that is a particular solution to the nonhomogeneous differential equation . Determine the general solution of the DE. Note: Any arbitrary constants used in the answer must be a lower-case "c".

    So far I've found the general solution without the particular solution to be:

    $\displaystyle y = t^3/5+c/t^2$

    Now what do I do with the particular solution? Any help is appreciated!
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  2. #2
    MHF Contributor chisigma's Avatar
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    Once You know a particular solution of the 'complete' DE, in order to solve the problem You have to find the general solution of the 'incomplete' DE that is...

    $\displaystyle t\cdot y^{'} - 2\cdot y=0$ (1)

    This linear DE is 'Euler type' and its solution has the form $\displaystyle y= c\cdot t^{\alpha}$. It is easy to verify that the only value of $\displaystyle \alpha$ that satisfies (1) is $\displaystyle \alpha=2$ so that the general solution of (1) is...

    $\displaystyle y=c\cdot t^{2}$ (2)

    ... and the general solution of the 'complete' DE is...

    $\displaystyle y=c\cdot t^{2} + t^{3}$ (3)

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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