Chapter 10 - Introduction to Differential Equations - 10.1 Solving Differential Equations - Exercises - Page 505: 27

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

$\frac{dy}{dx}=x\sec y$ Separating the variables, we have $\frac{dy}{\sec y}=xdx$ or $\cos y \, dy=xdx$ Integrating both sides, we get $\int\cos y\,dy=\int xdx$ $\sin y=\frac{x^{2}}{2}+C$ or $y=\sin^{-1}(\frac{x^{2}}{2}+C)$ where C is an arbitrary constant.