Prove the identity $\displaystyle \int_{0}^{x} \exp\left(zx - z^2\right) dz = \exp\left(\frac{x^2}{4}\right)\int_{0}^{x} \exp\left(\frac{-z^2}{4}\right) dz$ deriving for the function $\displaystyle I(x) = \int_{0}^{x} \exp\left(zx - z^2\right) dz$ as a differential equation and solving it.