The equation under consideration is

(d²φ/dx²) = −k²(x)φ(x)

where k(x) is a positive but declining function of x. This can be termed a generalized Helmholtz equation.

Now consider the points of relative maxima and minima of φ², labeled as x_i and y_i, respectively. These are points such that dφ²/dx=2φ(dφ/dx) equals zero; (dφ/dx)=0 for the maxima and φ=0 for the minima. Let the labeling be such that x_i > y_i.

Multiply the equation by (dφ/dx). to get

(dφ/dx)(d²φ/dx²) = −k²(x)φ(x)(dφ/dx)
which is the same as
½(d((dφ/dx)²)/dx) = −½k²(x)(d(φ²)/dx)

Now integrate the above equation from x=y_i to x=x_i. The result may be represented as

½[((dφ/dx)²]_yi^x_i = −½∫_yi^x_ik²(x)(φ²(x)dx

At x_i (dφ/dx) is equal to 0. By the Generalized Mean Value Theorem the RHS of the above equation may be expressed as

= −½∫_yi^x_ik²(x)(φ²(x)dx = −½k²(z_i)∫_yi^x_i(φ²(x)dx
which reduces to
−½k²(z_i)[φ²]_yi^x_i

where z_i is some point between y_i and x_i.

But φ² is equal to zero at y_i.

Therefore the previous equation reduces to

−½(dφ/dx)²_yi = −½k²(z_i)φ²(x_i)
or, equivalently
φ²(x_i) = (dφ/dx)²_yi/k²(z_i)

The average value of φ² over the interval [y_i, x_i] is essentially one half of the maximum value for that interval, which is its value at x_i. The maximum value of (dφ/dx)² occurs where d(dφ/dx)/dx is equal to zero. Since d(dφ/dx)/dx=−k²(x)φ(x) that maximum occurs where φ(x)=0. Thus the average of (dφ/dx)² over the interval [y_i, x_i] is also essentially one half of its maximum value of that interval, which is its value at y_i. The value of k²(z_i) is also its average value over the interval [y_i, x_i]. Let the average value of a variable be denoted by the underscore of its symbol. Thus

φ² = (dφ/dx)²/k²