2.1.12.1 Existence and uniqueness analysis

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

This is non linear first order ODE. In canonical form it is written as

\begin{align*} x^{\prime } &= f(t,x)\\ &= \tan \left (x \right ) \end{align*}

The \(x\) domain of \(f(t,x)\) when \(t=0\) is

\begin{align*} \left \{x <\frac {1}{2} \pi +\pi \_Z244 \boldsymbol {\lor }\frac {1}{2} \pi +\pi \_Z244 <x\right \} \end{align*}

And the point \(x_0 = 1\) is inside this domain. Now we will look at the continuity of

\begin{align*} \frac {\partial f}{\partial x} &= \frac {\partial }{\partial x}\left (\tan \left (x \right )\right ) \\ &= 1+\tan \left (x \right )^{2} \end{align*}

The \(x\) domain of \(\frac {\partial f}{\partial x}\) when \(t=0\) is

\begin{align*} \left \{x <\frac {1}{2} \pi +\pi \_Z244 \boldsymbol {\lor }\frac {1}{2} \pi +\pi \_Z244 <x\right \} \end{align*}

And the point \(x_0 = 1\) is inside this domain. Therefore solution exists and is unique.