﻿ The Solution to a Generalized Helmholtz Equation of One Dimension
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 The Solution to a Generalized Helmholtz Equation of One Dimension

The equation of interest is

Let

#### φ=ψ·exp(iζ) where ζ(s)=∫0sg(z)dz

where g(z) is a function yet to be determined.

Thus

Therefore

#### (d²ψ/ds²)exp(iζ) + 2(dψ/ds)exp(iζ)(ig(s)) − ψ·exp(iζ)g²(s) + ψ·exp(iζ)i(dg/ds)) = −k²ψ·exp(ζ) which, upon division by exp(iζ), reduces to (d²ψ/ds²) + 2(dψ/ds)ig(s) −ψg²(s) + ψi(dg/ds)) = −k²(s)ψ

Let g(z) be such that

#### −ψg²(s) + ψi(dg/ds)) = −k²(s)ψ or, equivalently (dg/ds) = i(k² − g²)

This is a nonlinear ordinary differential equation of the first order. At this point we only need to know that it has a solution. The name for this type of equation is Ricatti.

Thus the previous equation further reduces to

#### (d²ψ/ds²) + 2i(dψ/ds)g(s) = 0

Now let ν equal (dψ/ds). The above equation is then equivalent to

#### (dν/ds) = −2ig(s)ν or, equivalently (1/ν)(dν/ds) = −2ig(s) which is equivalent to (d(ln(ν)/ds) = −2ig(s)

This last equation has as its solution

This means that

And finally

#### φ(s) = ψ(s)·exp(iζ(s))

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