Let $y=y(x)$ be the solution curve of the differential equation $\sec y \frac{\mathrm{d} y}{\mathrm{~d} x}+2…
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 :
$\frac{\pi}{3}$
$\frac{\pi}{6}$
$\frac{\pi}{12}$
$\frac{\pi}{4}$
Solution
$\begin{aligned} & \sec ^2 y \frac{d y}{d x}+2 x \sin y \text { secy }=x^3 \cos y \text { secy } \\ & \sec ^2 y \frac{d y}{d x}+2 x \tan y=x^3 \\ & \operatorname{tany}=t \Rightarrow \sec ^2 y \frac{d y}{d x}=\frac{d t}{d x} \\ & \frac{d t}{d x}+2 x t=x^3, \text { If }=e^{\int 2 x d x}=e^{x^2} \\ & t x^{x^2}=\int x^3 \cdot e^{x^2} d x+c \\ & x^2=Z \Rightarrow t \cdot e^Z=\frac{1}{2} \int e^Z \cdot Z d Z=\frac{1}{2}\left[e^Z \cdot Z-e^Z\right]+c \\ & 2 \tan y=\left(x^2-1\right)+2 e^{-x^2} \\ & y(1)=0 \Rightarrow c=0 \Rightarrow y(\sqrt{3})=\frac{\pi}{4}\end{aligned}$