Natasha Gill

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:
${y}^{\prime }+x\mathrm{sin}\left(2y\right)=x{e}^{-{x}^{2}}{\mathrm{cos}}^{2}\left(y\right)$
I tried to find a result with wolfram web page (free version) and I got this one:
$y=\mathrm{arctan}\left(\frac{1}{2}{e}^{-{x}^{2}}\left(c+{x}^{2}\right)\right)$
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.

Dakota Cunningham

Expert

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 ${\mathrm{cos}}^{2}y$ gives
$\frac{{y}^{\prime }}{{\mathrm{cos}}^{2}y}+2x\mathrm{tan}y=x{e}^{-{x}^{2}}$
which has the form
${f}^{\prime }\left(y\right){y}^{\prime }+2xf\left(y\right)=x{e}^{-{x}^{2}}$
which now is linear in $u=f\left(y\right)=\mathrm{tan}\left(y\right)$. For this linear equation, ${e}^{{x}^{2}}$ is an integrating factor which miraculously also simplifies the right side. After integrating you get
${e}^{{x}^{2}}\mathrm{tan}\left(y\left(x\right)\right)={\frac{12}{x}}^{2}+c$

Do you have a similar question?