Solve dx/dy+(cotx)y=2cscx

$$\frac { d x } { d y } + ( \cot x ) y = 2 \csc x$$

Solve for x

$$x=ArcCosI(2y^{-1})+2\pi n_{9}\text{, }n_{9}\in \mathrm{Z}\text{, }\left(not(y<2)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(y>2\left(SinI(\frac{1}{2}\pi \left(1+\left(-2\right)n_{4}\right))\right)^{-1}\text{ and }not(n_{4}>1+2n_{9})\text{ and }not(n_{4}<-1+2n_{9})\text{ and }\exists n_{15}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{15}\right)\right)\text{ or }\left(not(|y|<2)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(y>2\left(SinI(\frac{1}{2}\pi \left(1+\left(-2\right)n_{4}\right))\right)^{-1}\text{ and }not(n_{4}>1+2n_{9})\text{ and }y<2\left(SinI(\left(-\frac{1}{2}\right)\pi +\left(-1\right)n_{4}\pi )\right)^{-1}\text{ and }not(n_{4}<-1+2n_{9})\text{ and }\exists n_{15}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{15}\right)\right)\text{ or }\left(not(|y|<2)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(y>2\left(SinI(\frac{1}{2}\pi \left(1+\left(-2\right)n_{4}\right))\right)^{-1}\text{ and }not(n_{4}>1+2n_{9})\text{ and }y>2\left(SinI(\left(-\frac{1}{2}\right)\pi +\left(-1\right)n_{4}\pi )\right)^{-1}\text{ and }not(n_{4}<-1+2n_{9})\text{ and }\exists n_{23}\in \mathrm{Z}\text{ : }n_{4}=-1+\left(-2\right)n_{23}\text{ and }\exists n_{15}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{15}\right)\right)\text{ or }\left(not(|y|<2)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(y<2\left(SinI(\frac{1}{2}\pi \left(1+\left(-2\right)n_{4}\right))\right)^{-1}\text{ and }\exists n_{13}\in \mathrm{Z}\text{ : }n_{4}=1+\left(-2\right)n_{13}\text{ and }not(n_{4}>1+2n_{9})\text{ and }y<2\left(SinI(\left(-\frac{1}{2}\right)\pi +\left(-1\right)n_{4}\pi )\right)^{-1}\text{ and }not(n_{4}<-1+2n_{9})\text{ and }\exists n_{15}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{15}\right)\right)\text{ or }\left(not(|y|<2)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(y<2\left(SinI(\frac{1}{2}\pi \left(1+\left(-2\right)n_{4}\right))\right)^{-1}\text{ and }\exists n_{13}\in \mathrm{Z}\text{ : }n_{4}=1+\left(-2\right)n_{13}\text{ and }not(n_{4}>1+2n_{9})\text{ and }y>2\left(SinI(\left(-\frac{1}{2}\right)\pi +\left(-1\right)n_{4}\pi )\right)^{-1}\text{ and }not(n_{4}<-1+2n_{9})\text{ and }\exists n_{23}\in \mathrm{Z}\text{ : }n_{4}=-1+\left(-2\right)n_{23}\text{ and }\exists n_{15}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{15}\right)\right)\text{ or }\left(not(y>-2)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(not(n_{4}>1+2n_{9})\text{ and }y<2\left(SinI(\left(-\frac{1}{2}\right)\pi +\left(-1\right)n_{4}\pi )\right)^{-1}\text{ and }not(n_{4}<-1+2n_{9})\text{ and }\exists n_{15}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{15}\right)\right)\text{ or }\left(not(y<2)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(not(n_{4}>1+2n_{9})\text{ and }\exists n_{13}\in \mathrm{Z}\text{ : }n_{4}=1+\left(-2\right)n_{13}\text{ and }not(n_{4}<-1+2n_{9})\text{ and }\exists n_{15}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{15}\right)\right)\text{ or }\left(not(y>-2)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(not(n_{4}>1+2n_{9})\text{ and }not(n_{4}<-1+2n_{9})\text{ and }\exists n_{23}\in \mathrm{Z}\text{ : }n_{4}=-1+\left(-2\right)n_{23}\text{ and }\exists n_{15}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{15}\right)\right)$$

$$x=2n_{38}\pi +\left(-1\right)ArcCosI(2y^{-1})\text{, }n_{38}\in \mathrm{Z}\text{, }\exists n_{4}\in \mathrm{Z}\text{ : }\left(\left(n_{4}\text{bmod}2=1\text{ and }not(y>-2)\text{ and }n_{4}=\left(-1\right)\left(\left(-2\right)n_{38}+2\right)\right)\text{ or }\left(not(|y|<2)\text{ and }y>\left(-2\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }n_{4}=\left(-1\right)\left(\left(-2\right)n_{38}+2\right)\right)\text{ or }\left(y>\left(-2\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }not(y>-2)\text{ and }n_{4}=\left(-1\right)\left(\left(-2\right)n_{38}+2\right)\right)\text{ or }\left(n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }not(y<2)\text{ and }n_{4}=\left(-1\right)\left(\left(-2\right)n_{38}+2\right)\right)\text{ or }\left(not(|y|<2)\text{ and }y<\left(-2\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }n_{4}=\left(-1\right)\left(\left(-2\right)n_{38}+2\right)\right)\text{ or }\left(y<2\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }not(y<2)\text{ and }n_{4}=2n_{38}\right)\text{ or }\left(n_{4}\text{bmod}2=1\text{ and }not(y<2)\text{ and }n_{4}=2n_{38}\right)\text{ or }\left(n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{38})\text{ and }y>\left(-2\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{38}\right))\text{ and }not(y<2)\right)\text{ or }\left(y<2\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }not(|y|<2)\text{ and }n_{4}=2n_{38}\right)\text{ or }\left(y<2\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{38})\text{ and }not(|y|<2)\text{ and }y<\left(-2\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{38}\right))\right)\text{ or }\left(y<2\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{38})\text{ and }not(|y|<2)\text{ and }y>\left(-2\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{38}\right))\right)\text{ or }\left(y<2\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{38})\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{38}\right))\text{ and }not(y>-2)\right)\text{ or }\left(y>2\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }not(|y|<2)\text{ and }n_{4}=2n_{38}\right)\text{ or }\left(y>2\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{38})\text{ and }not(|y|<2)\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{38}\right))\text{ and }not(y=\left(-2\right)\left(-1\right)^{\left(-1\right)n_{4}})\right)\text{ or }\left(n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }not(y>-2)\text{ and }n_{4}=2n_{38}\right)\text{ or }\left(n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{38})\text{ and }not(y>-2)\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{38}\right))\right)\text{ or }\left(n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{38})\text{ and }not(y<2)\text{ and }n_{4}\text{bmod}2=0\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{38}\right))\right)\right)$$

Solve for y

$$y=\frac{2}{\cos(x)}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\frac{\pi n_{1}}{2}$$