Trigonometric polynomial solutions of Bernouilli trigonometric polynomial differential equations

We consider real trigonometric polynomial Bernouilli equations of the form A(theta) y ' = B-1(theta) + B-n(theta)y(n) where n >= 2, with A, B-1, B-n being trigonometric polynomials of degree at most mu >= 1 in the variable. and B-n(theta) not equivalent to 0. We also consider the trigonometric polynomials of the form A(theta) y(n-1) y ' = B-0(theta) + B-n(theta) y(n) where n >= 2, being A, B-0, B-n trigonometric polynomials of degree at most mu >= 1 in the variable theta and B-n(theta) not equivalent to 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 above mentioned two types where the maximum number of trigonometric polynomial solutions is achieved. The method of proof will be applying extended Fermat problems for polynomial equations.