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

φ² = (dφ/dx)²/k²

Another Relationship

Consider again the generalized Helmholtz equation

(d²φ/dx²) = −k²(x)φ(x)

Now multiply both sides of this equation by (dφ/dx) to obtain

(dφ/dx)(d²φ/dx²) = −k²(x)φ(x)(dφ/dx)
which can be represented as
d(½(dφ/dx)²/dx) = −k²d(½φ²)/dx)

Now integrate the above equation from a to b to obtain

[½(dφ/dx)²]_{a}^{b} = −½∫_{a}^{b}k²(x)(φ(x)²(x)dx

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