I have been working with a system which involves computing the roots of functions that look like

\begin{equation}
e^t (g\cos(\omega t) + b) = c
\end{equation}

where $t$ is the independent variable and the parameters $g$, $\omega$ and $b$ are real and positive and $c$ is real. 

This, apparently has no analytical solution and I have been doing it in the computer easily. However, for some parts of my research it would be very useful to have some approx. solution. For instance, in the form of a series where successive approx. may be computed. However, I have no idea how to even start. Would anyone Have some guidelines as to how to approach this problem?

Thanks very much in advance. 

Edit: 

So, the full problem is as follows. Suppose we start with some value $t\_0$. Then we want the first root of 
\begin{equation}
e^{t\_1} (g\cos(\omega t\_1) + b) = e^{t\_0} (g\cos(\omega t\_0) + b)
\end{equation}

ie, the value of $t\_1$ that satisfies this equation and is closer to $t\_0$. Note that the RHS is simply the number which I previously called $c$. With $t\_1$, the next step is to compute $t\_2$ from the equation

\begin{equation}
e^{t\_2} (g\cos(\omega t\_2) - b) = e^{t\_1} (g\cos(\omega t\_1) - b)
\end{equation}
(note the sign change in $b$). 

This is the framework of my current problem: find successive roots $t\_i$ with the sign in $b$ alternating between each computation. 

The questions that arise are: for what values of $t\_0$ are there roots and are there infinitely many? For instance, if $\omega=3$, $b=0.8$ and $g=1.75$ then $t\_0 = 0.174765$ is unstable. It yields roughly $20$ roots and then stop. 

---

Update
------

Thank you all for the support. As for the instability (cf. @LeonidKovalev's answer), here is an example:

![enter image description here](https://i.stack.imgur.com/i2jon.png) 

This is for $\omega = 3$, $g= 1.75$ and $b = 0.8$. This shows that as time progresses the crossing times begin to grow so that after some time there is no longer a root. Perhaps, and this is a mistake which I apologise for, this instability is brought about by the fact that I am solving a slightly different system in which $\cos\omega t$ is replaced by $\cos(\omega t-\phi)$, with $\phi = \text{atan}(\omega)$ (I thought this would go away by rescaling $t$ appropriately).