Description
We present a systematic construction of a family of second-order linear ODEs of the form
y′′ + r(z)y = 0 whose solutions are Liouvillian and arise from a logarithmic derivative s(z) ∈
C(z, √R) of algebraic degree exactly 2 over C(z). The construction places two poles at z = 0
and z = 1 with equal residues 1/4, yielding the potential r(z) = 3
16z2(z−1)2 and solutions
expressible in terms of radicals and inverse hyperbolic functions. We verify the Riccati
equation by direct computation, prove that the logarithmic derivative has algebraic degree
2 via its minimal polynomial, and establish linear independence of the two fundamental
solutions via their nonconstant ratio and constant Wronskian. An outline of an n-pole
generalization is included. We have not found this exact inverse two-pole construction in
the standard references consulted.
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