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

Natasha Gill

Natasha Gill

Answered

2022-01-27

How To Solve a Trigonometric Differential Equation
Salutations, I have been trying to approach an ODE with trigonometric functions that I found interesting:
y+xsin(2y)=xex2cos2(y)
I tried to find a result with wolfram web page (free version) and I got this one:
y=arctan(12ex2(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.

Answer & Explanation

Dakota Cunningham

Dakota Cunningham

Expert

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
ycos2y+2xtany=xex2
which has the form
f(y)y+2xf(y)=xex2
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
ex2tan(y(x))=12x2+c

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