DSolve[y[x]*((g^3)[x]/(2*Derivative[1][g][x]) + Derivative[1][g][x]^2 + ((1/4 - v^2)*Derivative[1][g][x]^2)/g[x] - (3*Derivative[2][g][x]^2)/(4*Derivative[1][g][x]^2)) + Derivative[2][y][x] == 0,y[x],x]