Solve y^prime-ytanx=cosx

$$y ^ { \prime } - y \tan x = \cos x$$

Solve for y (complex solution)

$$y=-\frac{\cos(2x)+1}{2\sin(x)}$$

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

Solve for x

$$x=\pi +2n_{96}\pi +\left(-1\right)arcSin(\frac{1}{2}y+\left(-\frac{1}{2}\right)\left(y^{2}+4\right)^{\frac{1}{2}})\text{, }n_{96}\in \mathrm{Z}\text{, }\exists n_{91}\in \mathrm{Z}\text{ : }\left(\left(y>\left(-\frac{1}{4}\right)\left(4+\left(-4\right)\left(-1\right)^{n_{91}}SinI(\frac{1}{2}\pi \left(1+\left(-2\right)n_{91}\right))\right)\left(SinI(\frac{1}{2}\pi \left(1+\left(-2\right)n_{91}\right))\right)^{-1}\text{ and }\exists n_{100}\in \mathrm{Z}\text{ : }n_{91}=\left(-2\right)n_{100}\text{ and }not(n_{91}>1+2n_{96})\text{ and }n_{91}\text{bmod}2=0\text{ and }not(y<\left(-2\right)\left(-1\right)^{n_{91}})\text{ and }not(n_{91}<-1+2n_{96})\text{ and }y>0\right)\text{ or }\left(n_{91}\text{bmod}2=0\text{ and }not(y<\left(-2\right)\left(-1\right)^{n_{91}})\text{ and }n_{91}=-1+2n_{96}\text{ and }y>0\right)\text{ or }\left(not(y>2SinI(\frac{1}{2}\pi \left(1+\left(-2\right)n_{91}\right)))\text{ and }not(y<0)\text{ and }n_{91}=1+2n_{96}\right)\text{ or }\left(not(y>2SinI(\frac{1}{2}\pi \left(1+\left(-2\right)n_{91}\right)))\text{ and }not(n_{91}>1+2n_{96})\text{ and }n_{91}\text{bmod}2=0\text{ and }not(y<\left(-2\right)\left(-1\right)^{n_{91}})\text{ and }not(n_{91}<-1+2n_{96})\text{ and }y>0\right)\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2n_{96}\pi +\left(-1\right)arcSin(\frac{1}{2}y+\left(-\frac{1}{2}\right)\left(y^{2}+4\right)^{\frac{1}{2}})=\frac{1}{2}\pi +\pi n_{1}$$

$$x=arcSin(\frac{1}{2}y+\left(-\frac{1}{2}\right)\left(y^{2}+4\right)^{\frac{1}{2}})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }arcSin(\frac{1}{2}y+\left(-\frac{1}{2}\right)\left(y^{2}+4\right)^{\frac{1}{2}})+2\pi n_{2}=\frac{1}{2}\pi +\pi n_{1}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }arcSin(\frac{1}{2}y+\left(-\frac{1}{2}\right)\left(y^{2}+4\right)^{\frac{1}{2}})+2\pi n_{2}=\frac{1}{2}\pi +\pi n_{1}$$

$$x=\pi +2\pi n_{48}+\left(-1\right)arcSin(\frac{1}{2}y+\frac{1}{2}\left(y^{2}+4\right)^{\frac{1}{2}})\text{, }n_{48}\in \mathrm{Z}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2\pi n_{48}+\left(-1\right)arcSin(\frac{1}{2}y+\frac{1}{2}\left(y^{2}+4\right)^{\frac{1}{2}})=\frac{1}{2}\pi +\pi n_{1}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2\pi n_{48}+\left(-1\right)arcSin(\frac{1}{2}y+\frac{1}{2}\left(y^{2}+4\right)^{\frac{1}{2}})=\frac{1}{2}\pi +\pi n_{1}$$

$$x=arcSin(\frac{1}{2}y+\frac{1}{2}\left(y^{2}+4\right)^{\frac{1}{2}})+2\pi n_{47}\text{, }n_{47}\in \mathrm{Z}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }arcSin(\frac{1}{2}y+\frac{1}{2}\left(y^{2}+4\right)^{\frac{1}{2}})+2\pi n_{47}=\frac{1}{2}\pi +\pi n_{1}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }arcSin(\frac{1}{2}y+\frac{1}{2}\left(y^{2}+4\right)^{\frac{1}{2}})+2\pi n_{47}=\frac{1}{2}\pi +\pi n_{1}$$

Solve for y

$$y=-\cos(x)\cot(x)$$

$$\exists n_{1}\in \mathrm{Z}\text{ : }\left(x>\frac{\pi n_{1}}{2}\text{ and }x<\frac{\pi n_{1}}{2}+\frac{\pi }{2}\right)$$