2.1.2.2 Solved using first_order_ode_quadrature
Entering first order ode quadrature solver
\[\begin {aligned} x^{\prime }&=b \,{\mathrm e}^{t}\\ x \left (1\right ) &= 0\\ \end {aligned}\]
Because the ODE has the form \(x^{\prime }=f(t)\), the solution requires only integration. Therefore
\begin{align*} dx &= \left (b \,{\mathrm e}^{t}\right ) \, dt\\ x &= \int { \left (b \,{\mathrm e}^{t}\right ) \, dt}\\ &= b \,{\mathrm e}^{t}+c_1 \end{align*}
Applying the initial condition \(x \left (1\right ) = 0\), the solution becomes
\begin{align*}
x &= b \,{\mathrm e}^{t}-b \,{\mathrm e} \\
\end{align*}
Summary of solutions found
\[
x = b \,{\mathrm e}^{t}-b \,{\mathrm e}
\]