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Math Help - Integral equations

  1. #1
    Senior Member Twig's Avatar
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    Integral equations

    Hi!

    Problem: Find all continous solutions to y(x)-2-\int_{0}^{x} \; e^{y(t)-t} \; dt =0 and give their domains.


    y'(x)-e^{y(x)-x}=0 (, according to fundamental theorem of calculus)

    y'(x)-\frac{e^{y(x)}}{e^{x}} =0

    Can I separate the variables like  \frac{y'(x)}{e^{y(x)}}=\frac{1}{e^{x}} So now I would need to integrate both sides, right?

    \int \frac{1}{e^{y(x)}} \; dy = \int \frac{1}{e^{x}} \; dx

     -e^{y(x)}=-e^{x}+C

    From here I solve for y(x) , and use the fact that y(0)=2

    This gave me  y(x)=-ln(e^{-x}-1+e^{-2}) .

    For the domain, is this  x \mbox{ such that  } e^{-x}-1+e^{-2} >0 ?

    That would give  x<-ln(1-e^{-2})

    Kinda unsure about this though.

    Thanks!
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  2. #2
    MHF Contributor
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    Hi

    It seems OK to me, I would have done the same

    Just a typo here :  -e^{-y(x)}=-e^{-x}+C
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  3. #3
    Senior Member Twig's Avatar
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    Thank you
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