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Density Function Derived from the Solution to a Generalized Helmholtz Equation of One Dimension |
The equation under consideration is
where k(x) is a positive but declining function of x. This can be termed a generalized Helmholtz equation.
Now consider two adjacent points of relative maxima of φ², a and b. These are points such that dφ²/dx equal zero.
Multiply the equation by φ. and integrate the resulting equation from x=a to x=b. The result may be represented as
The LHS may be integrated by parts to yield
The quantity φ(dφ/dx) is equal to zero at both x=a and x=b. Therefore
Let K equal ∫_{a}^{b}k²(x)dx. Then dividing the above equation by K gives
Now divide the numerator and denominator on the RHS by (b−a) to obtain
The expression on the LHS of the above is a weighted average of φ². The expression on the RHS is the ratio of two "unweighted" (equal weighted) averages.
Thus the above equation can be represented as
Consider again the generalized Helmholtz equation
Now multiply both sides of this equation by (dφ/dx) to obtain
Now integrate the above equation from a to b to obtain
The expression on the LHS if zero because (dφ/dx) is zero both for x=a and x=b. The expression on the RHS of the above can be integrated by parts to obtain
This above equation can be expressed in the form
If Φ(x) is defined as k(x)²φ(x)² then the above equation takes the simple form
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