By using a general approach existing in the literature for distributions with density function satisfying a certain differential equation, a new bound is established for the solution of the Stein equation related to the generalized inverse Gaussian (resp. the Kummer) distribution. This bound is optimal for Lipschitz test functions. It has an explicit expression as a function of the parameters of the distribution in terms of the modified Bessel function of the third kind (resp. the confluent hypergeometric function of the second kind).