Exponentially small expansions related to the parabolic cylinder function

The refined asymptotic expansion of the confluent hypergeometric function $M(a,b,z)$ on the Stokes line $\arg\,z=\pi$ given in {\it Appl. Math. Sci.} {\bf 7} (2013) 6601--6609 is employed to derive the correct exponentially small contribution to the asymptotic expansion for the even and odd solutions of a second-order differential equation related to Weber's equation. It is demonstrated that the standard asymptotics of the parabolic cylinder function $U(a,z)$ yield an incorrect exponentially small contribution to these solutions. Numerical results verifying the accuracy of the new expansions are given.