Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations

Table 2

Derivation of novel Bessel and Euler equations.

Derived equations	Addition of Bessel equations
Semispherical Bessel	Regular and spherical
Modified semispherical Bessel	Modified and modified spherical
Regular Euler	Regular and modified
Spherical Euler	Spherical and modified spherical
Semispherical Euler	Regular and modified spherical
Semispherical Euler	Modified and spherical

Novel equations derived by using the binary operation of S.