# If y- et tane Prove that cause de 2y +22=0 0 OG

$y=x+\mathrm{tan}x\phantom{\rule{0ex}{0ex}}\frac{dy}{dx}=1+se{c}^{2}\phantom{\rule{0ex}{0ex}}\frac{{d}^{2}y}{d{x}^{2}}=2secx\xb7secx\mathrm{tan}x=2se{c}^{2}x\mathrm{tan}x\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{d}^{2}y}{d{x}^{2}}=\frac{2\mathrm{tan}x}{{\mathrm{cos}}^{2}x}\phantom{\rule{0ex}{0ex}}\Rightarrow {\mathrm{cos}}^{2}x\frac{{d}^{2}}{d{x}^{2}}=2\mathrm{tan}x\phantom{\rule{0ex}{0ex}}\Rightarrow {\mathrm{cos}}^{2}x\frac{{d}^{2}y}{d{x}^{2}}=2(y-x)\phantom{\rule{0ex}{0ex}}\mathbf{\Rightarrow}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2}}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{=}\mathbf{0}\phantom{\rule{0ex}{0ex}}\mathit{H}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{,}\mathbf{}\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{d}\mathbf{.}$