Prove identity using DEQ?

**fardeen_gen** · May 3rd 2009, 08:16 PM

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.

**Calculus26** · May 5th 2009, 03:20 AM

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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