What have you tried so far?
We already know the wave functions and energy levels when $\lambda=0$, they're Hermite functions and integers + a half, respectively. If you know that the parameter $\lambda$ determining the anharmonicity is small, you could use perturbation theory to expand the eigenfunctions and energy levels in a power series in $\lambda$, e.g.
$\psi = \pi^{-1/4}e^{-x^2/2}+\lambda\psi_1+\lambda^2\psi_2+\ldots$,
$E = \frac{1}{2}+\lambda E_1+\lambda_2 E_2+\ldots$
I'm happy to elaborate, but the wiki article on quantum mechanical perturbation theory has a better explanation than I could give. It also uses the anharmonic oscillator as an example, although just to find the ground-state energy. The Landau & Lifshitz book has a pretty good chapter on perturbation theory, as does Baym's book.
Alternatively, it's probably just a hypergeometric function, because experience has demonstrated that everything is some kind of hypergeometric function.