Trigonometric Polynomial Solutions of Bernoulli Trigonometric Polynomial Differential Equations

Abstract

We consider real trigonometric polynomial Bernoulli equations of the form A(θ)y′=B1(θ)+Bn(θ)yn where n≥2, with A,B1,Bn being trigonometric polynomials of degree at most μ≥1 in variables θ and Bn(θ)≢0. We also consider trigonometric polynomials of the form A(θ)yn−1y′=B0(θ)+Bn(θ)yn where n≥2, being A,B0,Bn trigonometric polynomials of degree at most μ≥1 in the variable θ and Bn(θ)≢0. For the first equation, we show that when n≥4, it has at most 3 real trigonometric polynomial solutions when n is even and 5 real trigonometric polynomial solutions when n is odd. For the second equation, we show that when n≥3, it has at most 3 real trigonometric polynomial solutions when n is odd and 5 real trigonometric polynomial solutions when n is even. We also provide trigonometric polynomial equations of the two types mentioned above where the maximum number of trigonometric polynomial solutions is achieved. The proof method will be to apply extended Fermat problems to polynomial equations.