حل dy/dx+ytanx=secx

$$\frac { d y } { d x } + y \tan x = \sec x$$

حل مسائل x

$$x=arcSin(y^{-1})+2\pi n_{9}\text{, }n_{9}\in \mathrm{Z}\text{, }\exists n_{4}\in \mathrm{Z}\text{ : }\left(\left(n_{4}\text{bmod}2=1\text{ and }not(y>-1)\text{ and }n_{4}=\left(-1\right)\left(2+\left(-2\right)n_{9}\right)\right)\text{ or }\left(not(|y|<1)\text{ and }y>\left(-1\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(2+\left(-2\right)n_{9}\right)\right)\text{ or }\left(y>\left(-1\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }not(y>-1)\text{ and }n_{4}=\left(-1\right)\left(2+\left(-2\right)n_{9}\right)\right)\text{ or }\left(n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }not(y<1)\text{ and }n_{4}=\left(-1\right)\left(2+\left(-2\right)n_{9}\right)\right)\text{ or }\left(not(|y|<1)\text{ and }y<\left(-1\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(2+\left(-2\right)n_{9}\right)\right)\text{ or }\left(y<\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }not(y<1)\text{ and }n_{4}=2n_{9}\right)\text{ or }\left(n_{4}\text{bmod}2=1\text{ and }not(y<1)\text{ and }n_{4}=2n_{9}\right)\text{ or }\left(n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{9})\text{ and }y>\left(-1\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{9}\right))\text{ and }not(y<1)\right)\text{ or }\left(y<\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }not(|y|<1)\text{ and }n_{4}=2n_{9}\right)\text{ or }\left(y<\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{9})\text{ and }not(|y|<1)\text{ and }y<\left(-1\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_{9}\right))\right)\text{ or }\left(y<\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{9})\text{ and }not(|y|<1)\text{ and }y>\left(-1\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{9}\right))\right)\text{ or }\left(y<\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{9})\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{9}\right))\text{ and }not(y>-1)\right)\text{ or }\left(y>\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|<1)\text{ and }n_{4}=2n_{9}\right)\text{ or }\left(y>\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_{9})\text{ and }not(|y|<1)\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{9}\right))\text{ and }not(y=\left(-1\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>-1)\text{ and }n_{4}=2n_{9}\right)\text{ or }\left(n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{9})\text{ and }not(y>-1)\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{9}\right))\right)\text{ or }\left(n_{4}\text{bmod}2=1\text{ and }not(n_{4}>2n_{9})\text{ and }not(y<1)\text{ and }n_{4}\text{bmod}2=0\text{ and }not(n_{4}<\left(-1\right)\left(2+\left(-2\right)n_{9}\right))\right)\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }arcSin(y^{-1})+2\pi n_{9}=\frac{1}{2}\pi +\pi n_{1}$$

$$x=\pi +2n_{18}\pi +\left(-1\right)arcSin(y^{-1})\text{, }n_{18}\in \mathrm{Z}\text{, }\exists n_{4}\in \mathrm{Z}\text{ : }\left(\left(\left(not(y<1)\text{ and }not(n_{4}<-1+2n_{18})\text{ and }n_{4}\text{bmod}2=0\text{ and }\exists n_{24}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{24}\text{ and }\exists n_{22}\in \mathrm{Z}\text{ : }n_{4}=1+\left(-2\right)n_{22}\text{ and }not(n_{4}>1+2n_{18})\right)\text{ or }\left(y<\left(SinI(\frac{1}{2}\pi \left(\left(-2\right)n_{4}+1\right))\right)^{-1}\text{ and }not(n_{4}<-1+2n_{18})\text{ and }n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }\exists n_{22}\in \mathrm{Z}\text{ : }n_{4}=1+\left(-2\right)n_{22}\text{ and }\exists n_{24}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{24}\text{ and }not(n_{4}>1+2n_{18})\text{ and }\nexists n_{26}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{26}\right)\text{ or }\left(not(y>-1)\text{ and }\left(-1\right)n_{4}\text{bmod}2=1\text{ and }n_{4}\text{bmod}2=0\text{ and }\exists n_{24}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{24}\text{ and }not(n_{4}>1+2n_{18})\text{ and }not(n_{4}<-1+2n_{18})\right)\text{ or }\left(not(y>-1)\text{ and }-1<\left(-1\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }not(n_{4}>1+2n_{18})\text{ and }\exists n_{22}\in \mathrm{Z}\text{ : }n_{4}=1+\left(-2\right)n_{22}\text{ and }\exists n_{26}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{26}\text{ and }n_{4}\text{bmod}2=0\text{ and }not(n_{4}<-1+2n_{18})\right)\text{ or }\left(not(y>-1)\text{ and }not(n_{4}<-1+2n_{18})\text{ and }n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }\exists n_{26}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{26}\text{ and }\exists n_{22}\in \mathrm{Z}\text{ : }n_{4}=1+\left(-2\right)n_{22}\text{ and }not(n_{4}>1+2n_{18})\right)\text{ or }\left(not(y>-1)\text{ and }not(n_{4}<-1+2n_{18})\text{ and }n_{4}\text{bmod}2=0\text{ and }n_{4}\text{bmod}2=1\text{ and }\exists n_{24}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{24}\text{ and }not(n_{4}>1+2n_{18})\right)\text{ or }\left(y<\left(-1\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }\exists n_{24}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{24}\text{ and }not(n_{4}>1+2n_{18})\text{ and }not(n_{4}<-1+2n_{18})\right)\text{ or }\left(y<\left(-1\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }not(n_{4}>1+2n_{18})\text{ and }\exists n_{22}\in \mathrm{Z}\text{ : }n_{4}=1+\left(-2\right)n_{22}\text{ and }\exists n_{26}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{26}\text{ and }not(n_{4}<-1+2n_{18})\right)\right)\text{ or }\left(\left(y>\left(SinI(\frac{1}{2}\pi \left(\left(-2\right)n_{4}+1\right))\right)^{-1}\text{ and }\exists n_{24}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{24}\text{ and }not(n_{4}>1+2n_{18})\text{ and }n_{4}\text{bmod}2=0\text{ and }not(n_{4}<-1+2n_{18})\text{ and }not(y<1)\right)\text{ or }\left(y>\left(SinI(\frac{1}{2}\pi \left(\left(-2\right)n_{4}+1\right))\right)^{-1}\text{ and }\exists n_{24}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{24}\text{ and }not(n_{4}>1+2n_{18})\text{ and }not(|y|<1)\text{ and }y<\left(-1\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=0\text{ and }not(n_{4}<-1+2n_{18})\right)\text{ or }\left(y>\left(SinI(\frac{1}{2}\pi \left(\left(-2\right)n_{4}+1\right))\right)^{-1}\text{ and }\exists n_{24}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{24}\text{ and }not(n_{4}>1+2n_{18})\text{ and }not(|y|<1)\text{ and }y>\left(-1\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }n_{4}\text{bmod}2=0\text{ and }not(n_{4}<-1+2n_{18})\right)\text{ or }\left(y>\left(SinI(\frac{1}{2}\pi \left(\left(-2\right)n_{4}+1\right))\right)^{-1}\text{ and }\exists n_{24}\in \mathrm{Z}\text{ : }n_{4}=\left(-2\right)n_{24}\text{ and }not(n_{4}>1+2n_{18})\text{ and }y<\left(-1\right)\left(-1\right)^{\left(-1\right)n_{4}}\text{ and }n_{4}\text{bmod}2=1\text{ and }not(n_{4}<-1+2n_{18})\text{ and }not(y<1)\right)\right)\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2n_{18}\pi +\left(-1\right)arcSin(y^{-1})=\frac{1}{2}\pi +\pi n_{1}$$

حل مسائل y

$$y=\frac{1}{\sin(x)}$$

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