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A Transformation of a Generalized Helmholtz Equation

The equation under consideration is

(d²φ/dx²) = −k²(x)φ

If k is constant then the solution is φ(x)=A·sin(kx)+B·cos(kx).
This suggests that the solution to the general equation may be a function of ∫_{0}^{x}k(ζ)dζ.

Consider a change of the independent variable such that

dz = k(x)dx
This means that
(dz/dx) = k(x)
and
z = ∫_{0}^{x}k(ζ)dζ

The first derivative of φ with respect to x is given by