We address finding solutions y ∈ 𝒞 2 ( ℝ + ) of the special (linear) ordinary differential equation x y ″ ( x ) + ( a x 2 + b ) y ′ ( x ) + ( c x + d ) y ( x ) = 0 for all x ∈ ℝ + , where a , b , c , d ∈ ℝ are constant parameters. This will be achieved in three special cases via separation and a power series method which is specified using difference equation techniques. Moreover, we will prove that our solutions are square integrable in a weighted sense—the weight function being similar to the Gaussian bell e − x 2 in the scenario of Hermite polynomials. Finally, we will discuss the physical relevance of our results, as the differential equation is also related to basic problems in quantum mechanics.