Abstract
The Lagrange equation is a quintic (fifth-degree) equation appearing in a stationary solution of the three-body problem in celestial mechanics. This equation has one positive root, which is the only required in the above problem. Here we apply an elementary real integral formula for the closed-form solution of nonlinear equations to the solution of the Lagrange equation. Two different approaches to this solution are described in detail and numerical results (obtained by both approaches) are displayed. Beyond the Lagrange equation, the present results are also applicable to the solution of other nontrivial quintic and higher-degree polynomial equations.
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