How To Solve a Trigonometric Differential EquationSalutations, I have been trying to approach an ODE with trigonometric functions that I found interesting:y′+xsin⁡(2y)=xe−x2cos2(y)I tried to find a result with wolfram web page (free version) and I got this one:y=arctan⁡(12e−x2(c+x2))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.

Question

Dakota Cunningham · Accepted Answer

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 givesy′cos2y+2xtan⁡y=xe−x2which has the formf′(y)y′+2xf(y)=xe−x2which 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 getex2tan⁡(y(x))=12x2+c