Question

$$\frac { d x } { d t } + x \tan t = 2 \sin t$$

Solve for t

$\left\{\begin{matrix}\\t=\pi n_{1}\text{, }n_{1}\in \mathrm{Z}\text{, }&\text{unconditionally}\\t=-\arccos(\frac{x}{2})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}\text{; }t=\arccos(\frac{x}{2})+2\pi n_{3}\text{, }n_{3}\in \mathrm{Z}\text{, }&x\neq 0\text{ and }x\leq 2\text{ and }x\geq -2\end{matrix}\right.$

Solve for x

$\left\{\begin{matrix}x=2\cos(t)\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }t=\pi n_{1}+\frac{\pi }{2}\\x\in \mathrm{R}\text{, }&\exists n_{2}\in \mathrm{Z}\text{ : }t=\pi n_{2}\end{matrix}\right.$