For unmanned aerial vehicle (UAV)-assisted wireless networks, the continuous trajectory designs generally suffer from infinite number of variables of the continuous UAV trajectory. In this paper, to avoid unexpected trajectory approximation and overcome the difficulty in obtaining an accurate continuous trajectory, we aim at characterizing an analytical optimal solution for jointly designing the resource allocation and continuous UAV trajectory. We focus on a scenario with UAV at a fixed altitude being deployed to assist the wireless communication with a ground user. With limited energy accessible for the wireless transmissions, we construct a throughput maximization problem for jointly optimizing the continuous transmit power and the UAV's continuous trajectory. Via duality analysis, we obtain the features of the optimal power control and successfully convert the dual problem to a series of pure trajectory design problems, which can be optimally addressed based on a mechanical equivalence approach. Afterwards, we accordingly propose an algorithm for optimally solving the dual problem, from which the optimal joint solution can be analytically constructed in a closed form. Finally, we also verify our proposed algorithm and confirm the optimality of the obtained solution via simulations.