2.1.8.1 Existence and uniqueness analysis

\[\begin {aligned} x^{\prime }&=b \,{\mathrm e}^{x}\\ x \left (0\right ) &= 1\\ \end {aligned}\]

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

\begin{align*} x^{\prime } &= f(t,x)\\ &= b \,{\mathrm e}^{x} \end{align*}

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

\begin{align*} \{-\infty <x <\infty \} \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 (b \,{\mathrm e}^{x}\right ) \\ &= b \,{\mathrm e}^{x} \end{align*}

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

\begin{align*} \{-\infty <x <\infty \} \end{align*}

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