Exact solutions of the Euler-Bernoulli equation for selected polynomially non-uniform beams used for acoustic black holes

In this work, we present a method for obtaining exact analytical solutions to the Euler-Bernoulli equation for nonuniform beams with continuously varying rectangular cross-sections. The approach is based on factorizing the fourth-order equation into a system of second-order differential equations with variable coefficients. Focusing on polynomial expressions for the cross-sectional profile, we show that such factorization is possible only when the profile is described by a polynomial of at most third order. In the general cubic case, the resulting equation transforms into Heun's differential equation; in degenerate cases, it reduces to the hypergeometric or Bessel equations, all of which admit closed-form solutions. To demonstrate the method's applicability, we compute reflection coefficients for selected profiles relevant to Acoustic Black Holes and validate the analytical results using a Riccati-based numerical method, showing excellent agreement.