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A Solution to the Generalized
Helmholtz Equation of One Dimension

The Helmholtz equation arises in many contexts in the attempt to give a mathematical explanation of the physical world.
These range from Alan Turing's explanation of animal coat patterns to Schrödinger's time-independent equation
in quantum theory.

The Helmholtz equation per se is

∇²φ = −k²φ

where k is a constant. The Generalized Helmholtz equation is that equation with k being a function of the independent variable(s).

The One Dimensional Case

In one dimension the Helmholtz equation is

(d²φ/dx²) = −k²φ(x)

It just has the sinusoidal solution of φ(x) = A·sin(kx)+B·cos(kx).
In one dimension the Generalized Helmholtz equation has a sinusoidal-like solution of varying amplitude and wavelength.

Change of Variable

The sinusoidal solution being a function of kx suggests that the solution at the generalized equation may a function
of

X=∫_{0}^{x}k(z)dz.
hence
dX=k(x)dx
and
(dX/dx) = k(x)