A decomposed Fourier series solution to Prandtl's classical lifting-line theory is used to predict the lift, induced-thrust, and power coefficients developed by a flapping wing. A significant advantage of this quasi-steady analytical solution over commonly used numerical methods is the utility provided for optimizing wing flapping cycles. The analytical solution involves five time-dependent functions that could all be optimized to maximize thrust, propulsive efficiency, and/or other performance measures. Results show that by optimizing only two of these five functions, propulsive efficiencies exceeding 97% can be obtained. Results are presented for untwisted rectangular wings in pure plunging, rectangular wings with linear washout and the minimum-power washout magnitude, and rectangular wings with the minimum-power washout distribution and magnitude.