# What is a solution to the differential equation dy/dx=1/sec^2y?

Oct 20, 2016

$y = {\tan}^{- 1} \left(x + C\right)$

#### Explanation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{\sec} ^ 2 y$

This is a first order separable Differential Equation; so we can "separate the variables" to give:

$\int {\sec}^{2} y \mathrm{dy} = \int 1 \mathrm{dx}$

Integrating gives:

$\tan y = x + C$, where $C$ is the constant of integration
Hence, the solution is:

$y = {\tan}^{- 1} \left(x + C\right)$

Oct 20, 2016

$y = \arctan \left(x + C\right)$

#### Explanation:

Grouping variables

$\frac{\mathrm{dy}}{\cos} ^ 2 y = \mathrm{dx}$ so

$\tan y = x + C$ so

$y = \arctan \left(x + C\right)$