Solve explicitly for time from a sum of exponentials

Question

Suppose $f(t) = 0$. How can I solve for time, $t$, in the following expression.

$f(t) = k_1{e}^{- \alpha t} + k_2{e}^{- \beta t} + k_3{e}^{- \gamma t}\left( k_4 sin(\omega_d t) + k_5 cos (\omega_d t) \right) $

Answer 1

For this kind of equations (which mix exponential and trigonometric functions), only numerical methods (such as Newton) could solve the problem (assuming that a reasonable estimate is available).

Assuming that the first root is "small", you could expand the function as a Taylor series at $t=0$, limit to the first order $$f(x)=({k_1}+{k_2}+{k_3} {k_5})+t (-\alpha {k_1}-\beta {k_2}+{k_3} {k_4} \omega -\gamma {k_3} {k_5})+O\left(t^2\right)$$ and solve for $t$. This is equivalent to the first iteration of Newton method starting at $t_0=0$ hoping that this is a reasonable starting guess.