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Math Help - Commet on General Solution

  1. #1
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    Commet on General Solution

    Comment on the statement: The general solution of \varphi''+6\varphi'+\lambda\varphi=0 is

    \varphi=C_1\exp(-3+i\sqrt{\lambda-9})+C_2\exp(-3-i\sqrt{\lambda-9}).

    I am not sure what this question wants me to comment about unless it is the fact it should be written like

    \varphi=e^{-3}\left[A\cos(x\sqrt{\lambda-9})+B\sin(x\sqrt{\lambda-9})\right]
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    When will the solution be a sum of exponential functions?
    When will the solution need to have an extra factor of \displaystyle x?
    When will your solution need to use sines and cosines?
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