2.1.8.2 Solved using first_order_ode_autonomous
Entering first order ode autonomous solver
\[\begin {aligned} x^{\prime }&=b \,{\mathrm e}^{x}\\ x \left (0\right ) &= 1\\ \end {aligned}\]
Integrating gives
\begin{align*} \int \frac {{\mathrm e}^{-x}}{b}d x &= dt\\ -\frac {{\mathrm e}^{-x}}{b}&= t +c_1 \end{align*}
Applying the initial condition \(x \left (0\right ) = 1\), the solution becomes
\begin{align*}
-\frac {{\mathrm e}^{-x}}{b} &= t -\frac {{\mathrm e}^{-1}}{b} \\
\end{align*}
Solving for \(x\) from \(-\frac {{\mathrm e}^{-x}}{b} = t -\frac {{\mathrm e}^{-1}}{b}\) gives
\begin{align*}
x &= -\ln \left (-\left (t b \,{\mathrm e}-1\right ) {\mathrm e}^{-1}\right ) \\
\end{align*}
Summary of solutions found
\[
x = -\ln \left (-\left (t b \,{\mathrm e}-1\right ) {\mathrm e}^{-1}\right )
\]