Finding Equation of Tangent parallel to x & y axis

The Schrödinger equation is but a differential equation, so, then, solvable by one of the most effective iterative approximation techniques for such, namely WP:Asymptotic analysis; WP:Asymptotic expansion; WP:Asymptotic theory; etc...

Asymptotic analysis consists of either starting from a base solution and improving it iteratively as a series, or else, more suitably, by inferring the asymptotic dominant balance, as outlined and illustrated in the first article. I'm not sure what type of physical interpretation you are seeking in a mere differential equation.

To appreciate the strict definition of asymptotics, consider the WP:Big O notation.