Abstract
We consider real trigonometric polynomial Bernouilli equations of the form \(A(\theta ) y' =B_1(\theta ) +B_n(\theta ) y^n\) where \(n \ge 2\), with \(A,B_1,B_n\) being trigonometric polynomials of degree at most \(\mu \ge 1\) in the variable \(\theta \) and \(B_n(\theta ) \not \equiv 0\). We also consider the trigonometric polynomials of the form \(A(\theta ) y^{n-1} y' =B_0(\theta ) + B_{n}(\theta ) y^n\) where \(n \ge 2\), being \(A,B_0,B_{n}\) trigonometric polynomials of degree at most \(\mu \ge 1\) in the variable \(\theta \) and \(B_{n}(\theta ) \not \equiv 0\). For the first equation we show that when \(n \ge 4\) 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 \(n \ge 3\) 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.
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30 January 2024
This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1007/s13324-024-00875-5
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The author has been partially supported by FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020.
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Valls, C. RETRACTED ARTICLE: Trigonometric polynomial solutions of Bernouilli trigonometric polynomial differential equations. Anal.Math.Phys. 13, 33 (2023). https://doi.org/10.1007/s13324-023-00798-7
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DOI: https://doi.org/10.1007/s13324-023-00798-7