Let y=y(x) be the solution curve of the differential equation \sec y \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x \sin y=x^3 \cos y, y(1)=0. Then y(\sqrt{3}) is equal to:

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Answer: D. \frac{\pi}{4} \begin{aligned} & \sec y \frac{d y}{d x}+2 x \sin y=x^3 \cos y \ & \Rightarrow \sec ^2 y \frac{d y}{d x}+2 x 	an y=x^3 \end{aligned} Let z=	an y \begin{aligned} & \frac{d z}{d x}=\sec ^2 y \frac{d y}{d x} \ & \Rightarrow \frac{d z}{d x}+2 x z=x^3 \end{aligned} \begin{aligned} & 	ext { I.F. }=e^{x^2} \ & \Rightarrow z . e^{x^2}=\int e^{x^2} \cdot x^3 d x+c \end{aligned} \Rightarrow 	an y \cdot e^{x^2}=\frac{1}{2}\left(x^2 e^{x^2}-e^{x^2}ight)+c \Rightarrow 	an (0) \cdot e=\frac{1}{2}(1 \cdot e-e)+c \Rightarrow c=0 \Rightarrow 	an y=\frac{x^2-1}{2} f(x)=	an ^{-1}\left(\frac{x^2-1}{2}ight) \Rightarrow f(\sqrt{3})=\frac{\pi}{4}

Let $$y=y(x)$$ be the solution curve of the differential equation $$\sec y \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x \sin y=x^3 \cos y, y(1)=0$$. Then $$y(\sqrt{3})$$ is equal to:

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