little engineering/little calc

In dealing with metal surfaces that undergo oxidation, the thickness of the oxide film has been found to increase exponentially with time, such that d=dmax (1-e^(-kt)), t>2000 where d=thickness of oxide film and t=time.

This expression is the result of integrating an ordinary differential equation.

I'm trying to derive the Ordinary differential equation and the associated initial condition needed that produces d=dmax (1-e^(-kt))

Any help for how to approach this is much appreciated!