Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations

Abstract

For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions.

Keywords

• ordinary differential equations
• analytical methods
• mathematical models
• Riccati equation
• radial Schrödinger equation
• transformations