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Math Help - Two (probably stupidly simple) differential equations questions

  1. #1
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    Talking Two (probably stupidly simple) differential equations questions

    Hello!

    I never thought that there would be a math forum that would be this busy! Bookmarked.

    Ok... enough with the sugar. I have finished almost all of my homework sheet, but I am having problems with a couple questions. My text (First course in DE, by Zill) is not helping me at all, and neither is the almighty google.

    The first is asking me to find a differential equation that is satisfied by the function y=(C1+C2x)e^x

    I'm guessing it has to do with superposition, but I only know how to use that to get a solution from a DE, not vice versa!

    Next up, I have a question asking me to find the intervals where these DEs have a unique solution:

    a) (x^2-1)y''+xy'-5y=3
    and
    b) (x^3-1)y''+3y=5

    EVERY question I've seen for finding intervals with a unique solution atleast has initial values.

    I have been , and of course, the math help centre at my school only has a DE assistant when I am at work, so any help you guys give me would be frickin awesome.

    Thanks!
    Last edited by mr fantastic; October 19th 2009 at 04:49 PM.
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  2. #2
    MHF Contributor chisigma's Avatar
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    These are not at all stupid questions!...

    First answer: the general solution of the DE is...

    y= c_{1}\cdot e^{x} + c_{2}\cdot x\cdot e^{x} (1)

    ... and that means that is a second order linear DE with constant coefficients and its 'characteristic equation' has solution d=1 with molteplicity two, i.e. is...

    d^{2} - 2\cdot d + 1=0 (2)

    The corresponding DE is...

    y^{''} -2\cdot y^{'} + y =0 (3)

    Second answer: the interval in which a linear second order DE has one and only one solution for any 'initial condition' is the inteval in which the 'coefficient' a(x) of the term y^{''} doesn't vanish, i.e.  a(x)\ne 0. In a) is a(x)= x^{2}-1, in b) is a(x)= x^{3}-1, so that...

    Kind regards

    \chi \sigma
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  3. #3
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    Thanks alot! That helped a lot. I can't believe how easy the second question was. It was worth as many marks as a question that took 5 times as long to do, so that is why I got tripped up by it.



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