Analytic approximate solution for a nonlinear oscillator equation

Analytical approximate oscillatory solutions for the differential equation u″ + ω2u = λum with the boundary conditions u(0) = 0, u(1) = α are derived using the Adomian decomposition method. The parameters ω, λ, and α are positive real numbers, and m is a positive integer. The solutions are given in the form of series with easily computable terms and are valid for all ω including the critical frequencies. The validity of the solutions is verified through some numerical examples.