Ok. So the problem is:
Show that the damped equation
$\ddot{x} + k\dot{x} + (\gamma + \beta cos(t))x = 0$
can be transformed into a Mathieu equation by the change of variable $x = ze^{\mu t}$ for a suitable choice for $\mu$
Second attempt at solution:
We have:
$x=ze^{\mu t}$
$\dot{x}=\mu z e^{\mu t} + \dot{z}e^{\mu t}$
$\ddot{x} = \mu^2 ze^{\mu t}+2\mu \dot{z}e^{\mu t} + \ddot{z} + e^{\mu t}$
Inserting this into our equation gives, after some changing around:
$\ddot{z}e^{\mu t} + (2\mu + k)\dot{z}e^{\mu t} + (\mu^2 + k\mu + \gamma + \beta cos(t))ze^{\mu t} = 0$
We then choose $\mu$ so that $2\mu + k = 0$. By doing this, and dividing both sides with $e^{\mu t}$ we obtain:
$\ddot{z} + (\mu^2 + k\mu + \gamma + \beta cos(t))z = 0$
And now we just perform a simple change of variable, as shown in my last post, and we are done!
Does it look OK now? :)