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Math Help - General solution of a transformed differential equation

  1. #1
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    General solution of a transformed differential equation

    I have a differential equation of tx''+x'+tx=0; the initial conditions are x(0)=1 and x'(0)=0.

    I perform the Laplace transformation and I obtain the expression:
    ((s^2)+1)*X'(s)+s*X(s)=0. I have verified this with my textbook.

    The problem is that the textbook then says that X'(s)/X(s) = -s/((s^2)+1) of which I understand but it suddenly jumps to conclude that
    X(s) = C/((s^2)+1)^(0.5) where C is non-zero.

    I have looked at a couple other examples and I see the pattern that the answer is generally X(s) = C/(coefficient of X'(s))^((coefficient of X(s))/2) if there's an 's' next to X(s) and simply C/(coefficient of X'(s))^((coefficient of X(s))).

    So what is the proper way to do this?
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  2. #2
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    Re: General solution of a transformed differential equation

    I believe what the book did was integrate both sides

    \int \dfrac{dX}{X} = - \int \frac{s}{s^2+1}\,ds

    so

    \ln X = -\frac{1}{2} \ln (s^2 + 1) + \ln C

    and solving for X gives the books answer.
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