$x'' + \delta x' -x + x^3 = \gamma \cos \omega t \tag 1$
is an example of Duffing's equation. No closed form solution is known, and I doubt it has one in terms of elementary functions. The solutions exhibit many important and engaging phenomena, ranging from bounded oscillation to chaos, depending on the parameter values. Check out the linked citing; it's worth reading.
Meanwhile, if what is wanted here is to simply cast (1) into first-order form, we can exploit the standard procedure of setting
$v = x', \tag 2$
$v' = x'', \tag 3$
and then (1) may be written
$v' + \delta v - x + x^3 = \gamma \cos \omega t, \tag 4$
$v' = - \delta v + x - x^3 + \gamma \cos \omega t, \tag 5$
which together with (2) forms the first order system
$x' = v, \tag 6$
$v' = - \delta v + x - x^3 + \gamma \cos \omega t \tag 7$
in the two variables $x$ and $v$.