The solution of the differential equation ${\cos }^{2}x\frac{d^{2}y}{d{x}^2}=1$ is:

     ${\cos }^{2}x\frac{d^{2}y}{d{x}^2}=1$

$\Rightarrow \frac{d^{2}y}{d{x}^2}={\sec }^{2}x$

On integrating, we get $\frac{dy}{dx}=\tan x\pm c_1$

Again integrating, we get $y=\log \sec x\pm c_{1}x\pm c_2.$