How To Solve a Trigonometric Differential EquationSalutations, I have been trying to approach an ODE...

Natasha Gill

Natasha Gill



How To Solve a Trigonometric Differential Equation
Salutations, I have been trying to approach an ODE with trigonometric functions that I found interesting:
I tried to find a result with wolfram web page (free version) and I got this one:
I have tried to approach this exercise by substitution of variables, also separable variables and I have not had luck by power series, and I do not know if methods like those of Ricatti and Bernoulli are appropriate for this case.

Answer & Explanation

Dakota Cunningham

Dakota Cunningham


2022-01-28Added 9 answers

This is in general so non-linear that you can not expect a closed solution. However, as it is an exercise a closed solution most probably exists, so you have to consider the parts of this equation. With some experience one may see that dividing by cos2y gives
which has the form
which now is linear in u=f(y)=tan(y). For this linear equation, ex2 is an integrating factor which miraculously also simplifies the right side. After integrating you get

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