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Thread: "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

    $\displaystyle y(t)=e^{mt},$

    and plug it into the equation. Derivatives become corresponding powers of $\displaystyle m,$ 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

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