The answer is yes. Indeed, let be that solution to the Prüfer d.e. which satisfies the initial condition . We have one such solution for each . We can draw the graphs of these solutions for various values of . See Figure 3.13.
Note that if the function of the S-L equation is the constant function, then the slope
at every ``zero'' of will be a fixed constant, independent of . However, between a pair of successive ``zeroes'' the slope will be the larger, the larger is. Consequently, for large , the phase has an undulatory (i.e., ``wavelike'') behavior as passes through the successive zeroes of .
We now ask: How does
behave as a function of ? The answer to this question is important because it determines whether the other end point condition ( , ) can be satisfied.
The behavior of as a function of is summarized by the following three statements, which together comprise the
The solution of the Prüfer d.e. satisfying the initial condition