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Math Help - Prove identity using DEQ?

  1. #1
    Super Member fardeen_gen's Avatar
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    Prove identity using DEQ?

    Prove the identity \int_{0}^{x} \exp\left(zx - z^2\right) dz = \exp\left(\frac{x^2}{4}\right)\int_{0}^{x} \exp\left(\frac{-z^2}{4}\right) dz deriving for the function I(x) = \int_{0}^{x} \exp\left(zx - z^2\right) dz as a differential equation and solving it.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    See the attachment

    Basically we start with



    We complete the square and use the FTC

    This leads to a linear DE we solve and obtain

    the right hand side of



    as a solution
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