On some conjectures of Turcotte, Spence, Bau, and Holmes

Turcotte, Spence, and Bau [Internat. J. Heat Mass Transfer, 25 (1982), pp. 699-706] contains conjectures concerning the equation \[ V'' = {{(V^2 - A(1 - X^2 ))} 2}\] with boundary conditions $V( - 1) = V(1) = 0$, where $A$ is a nonnegative parameter. For large $A$ an appropriate asymptotic expansion results in a version of the first Painleve transcendent, namely $Y'' = {{(Y^2 - s)} / 2}$ seeking a solution such that $Y(0) = 0$, $Y(s) - \sim \sqrt s $ as $s \to \infty $. This was studied extensively by Holmes and Spence, who conjectured that there are only two solutions. In this paper proofs of these conjectures are provided. During one of these proofs it is shown how a computer language for symbol manipulation, such as MACSYMA, can be used in a mathematically rigorous analysis.