Your differential equation is
deq = g[x] s''[x] + 2 g[x] s[x] + g''[x] s[x] == g[x]/2
Now make the following rearrangements
de1 = #/g[x] & /@ deq // Expand
de2 = # - 2 s[x] - s''[x] & /@ de1
de3 = #/s[x] & /@ de2
so the functions g and s are separated and you may set each side of the resulting equation de3 to an arbitrary constant or an arbitrary convenient function and solve the resulting differential equations.
For example
fg = g[x] /. DSolve[D[g[x], x, x]/g[x] == a, g[x], x][[1]]
fs = s[x] /. DSolve[(1/2 - 2 s[x] - D[s[x], x, x])/s[x] == a, s[x], x][[1, 1]]
and your differential equation again
2 fg fs + fs D[fg, x, x] + fg D[fs, x, x] - fg/2 // FullSimplify