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