2.3.4.1 Solved using first_order_ode_separable

Entering first order ode separable solver

\[\begin {aligned} \sec \left (x \right )^{2} \tan \left (y\right ) y^{\prime }+\sec \left (y\right )^{2} \tan \left (x \right )&=0 \end {aligned}\]

The ode

\begin{align*} y^{\prime } = -\frac {\sec \left (y\right )^{2} \tan \left (x \right )}{\sec \left (x \right )^{2} \tan \left (y\right )} \end{align*}

is separable as it can be written as

\begin{align*} y^{\prime }&= -\frac {\sec \left (y\right )^{2} \tan \left (x \right )}{\sec \left (x \right )^{2} \tan \left (y\right )}\\ &= f(x) g(y) \end{align*}

Where

\begin{align*} f(x) &= -\frac {\tan \left (x \right )}{\sec \left (x \right )^{2}}\\ g(y) &= \frac {\sec \left (y \right )^{2}}{\tan \left (y \right )} \end{align*}

Integrating gives

\begin{gather*} \begin {aligned} \int { \frac {1}{g(y)} \,dy} &= \int { f(x) \,dx}\\ \int { \frac {\tan \left (y \right )}{\sec \left (y \right )^{2}}\,dy} &= \int { -\frac {\tan \left (x \right )}{\sec \left (x \right )^{2}} \,dx} \end {aligned} \end{gather*}

Which becomes

\begin{align*} -\frac {\cos \left (y\right )^{2}}{2}&=\frac {\cos \left (x \right )^{2}}{2}+c_1 \end{align*}

Solving for \(y\) from \(-\frac {\cos \left (y\right )^{2}}{2} = \frac {\cos \left (x \right )^{2}}{2}+c_1\) gives

\begin{align*} y &= \arccos \left (\sqrt {-\cos \left (x \right )^{2}-2 c_1}\right ) \\ y &= \pi -\arccos \left (\sqrt {-\cos \left (x \right )^{2}-2 c_1}\right ) \\ \end{align*}

Direction field with Isoclines	Direction field

Summary of solutions found

\begin{align*} y&=\arccos \left (\sqrt {-\cos \left (x \right )^{2}-2 c_1}\right )\\ y&=\pi -\arccos \left (\sqrt {-\cos \left (x \right )^{2}-2 c_1}\right ) \end{align*}