Solved using ﬁrst_order_ode

Entering ﬁrst order ode autonomous solver

\[\begin {aligned} x^{\prime }&=\tan \left (x\right )\\ x \left (0\right ) &= 1\\ \end {aligned}\]

Integrating gives

\begin{align*} \int \frac {1}{\tan \left (x \right )}d x &= dt\\ \ln \left (\sin \left (x \right )\right )&= t +c_1 \end{align*}

Applying the initial condition \(x \left (0\right ) = 1\), the solution becomes

\begin{align*} \ln \left (\sin \left (x\right )\right ) &= t +\ln \left (\sin \left (1\right )\right ) \\ \end{align*}

Solving for \(x\) from \(\ln \left (\sin \left (x\right )\right ) = t +\ln \left (\sin \left (1\right )\right )\) gives

\begin{align*} x &= \arcsin \left (\sin \left (1\right ) {\mathrm e}^{t}\right ) \\ \end{align*}


Direction ﬁeld with Isoclines	\( x = \arcsin \left (\sin \left (1\right ) {\mathrm e}^{t}\right ) \)

Summary of solutions found

\[ x = \arcsin \left (\sin \left (1\right ) {\mathrm e}^{t}\right ) \]