The problem, as stated, is not physically realistic on a quantum level, given Earth's value of , because would have to be much too small. But an analogous experiment with a charge in an electric field is possibly more accessible. We will continue to refer to the gravitational parameters, however.

Redefining the independent variable as , the equation reduces to the simpler form . (The form of the variable is suggested by running DSolve on the original equation). The solution that remains finite as is found to be . (A second solution, , diverges as .)

The eigenvalues can be found from the zeros of the Airy function: , using N[AiryAiZero[n]]. The roots lie on the negative real axis, the first few being approximately , , , , , , …. Defining the constant , the lowest eigenvalues are thus given by , , , and so on. The corresponding (unnormalized) eigenfunctions are . These are plotted on the graphic.