# Thread: Prove identity using DEQ?

1. ## 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.

2. See the attachment

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