I am interested in a method to find the roots of the following equation:
\begin{equation} f(t) = \sum_{i=1}^n \alpha_i e^{\beta_i t} + \gamma t + \delta = 0. \end{equation}
For my application, coefficients $\alpha_i$, $\beta_i$, $\gamma$, and $\delta$ are real. $n$ is typically a small integer, say 10. In particular I am interested in the smallest positive real root of $f$.
For those interested, this equation arises when attempting to compute the point of intersection between the solution to the linear ODE
\begin{align} \dot x(t) &= Ax(t) + b & (A = A^T) \\ x(0) &= x_0 \end{align}
and the boundary of a set of linear constraints
\begin{equation} Cx(t) \ge d. \end{equation}
The initial point is always feasible $(Cx_0\ge d)$. For my purposes, all matrices and vectors are real.