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Math Help - "Match the third order linear equations with their fundamental solution sets"

  1. #1
    s3a
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    "Match the third order linear equations with their fundamental solution sets"

    The question is attached. How do I go about doing this? If someone does just one of them that should be enough for me to get it.

    Judging via y''' - 5y'' + 6y'= 0 - Wolfram|Alpha, the answer for #1 is C since the general solution has those 3 particular solutions but how do I do it without cheating?

    Any help would be greatly appreciated!
    Thanks in advance!
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  2. #2
    A Plied Mathematician
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    Re: "Match the third order linear equations with their fundamental solution sets"

    The key to all of those problems, except # 5, is to write the characteristic equation, which you obtain simply by assuming a solution of the form

    y(t)=e^{mt},

    and plug it into the equation. Derivatives become corresponding powers of m, and the exponentials always cancel because they're nonzero. You must then solve the resulting polynomial. Complex solutions you can resolve into trigonometric functions via the formula

    e^{it}=\cos(t)+i\sin(t).
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