Let $y_1(x)$ be an elementary solution of the Sturm-Liouville equation,
$$ \frac{d}{dx}\left( p(x)\frac{dy}{dx} \right) + q(x)y = 0 $$
It is well known that a second linearly independent solution may be computed from the reduction of order formula, provided that $y_1(x)$ is nonvanishing. I came upon a formula attributed to Fedor Rofe-Beketov which relaxes this constraint and claims to give a second linearly independent solution without any requirements as to the oscillation of $y_1(x)$:
$$ y_2(x) = y_1(x)\int \frac{(p(t)^{-1} - q(t))[(y_1(t)^2 - (p(t)y_1(t)')^2]}{[y_1(t)^2 + (p(t)y_1(t)')^2]^2} dt - \frac{p(x)y_1(x)}{y_1(x)^2 + (p(x)y_1(x)')^2} $$
which can be established by a somewhat lengthy but relatively straightforward computation. But in every example I can think of where a second solution must be determined when only one is known, it can be obtained by the ordinary reduction of order formula. For other problems, any algorithmic solution procedure should automatically gives both fundamental solutions and so neither formula is necessary. Perhaps this merely signifies a lack of imagination on my part, but are there any examples of applications of the Rofe-Beketov formula actually being required to complete the solution of an ODE?
The formula has the advantage over d'Alembert's more well-known formula that it gives an expression for the second solution that is unencumbered with the problems d'Alembert's formula has at zeros of the first solution. In my paper "Critical Coupling Constants and Eigenvalue Asymptotics of Perturbed Periodic Sturm-Liouville Operators", Commun. Math. Phys. 211 (2000) 465-485, Rofe-Beketov's formula is used in a situation where the first solution has infinitely many zeros, and then motivates a transformation that leads to the desired result; d'Alembert's formula will not get you anywhere in this problem. This paper also provides a reference to Rofe-Beketov's paper (who only considers the case p = 1). K. M. Schmidt