Integrate[(-1 - y'[x]^2 - 2 y[x] y''[x])/(
 2 y[x]^(3/2) (1 + y'[x]^2)^(3/2)), {y[x], 0, 1}]

solut = DSolve[{D[#, y[x]] - D[D[#, y'[x]], x] == 0}, y[x], x] &@ Sqrt[(1 + y'[x]^2)/(4.428 y[x])]
slt = ((y[x] /. solut) /. {C[1] -> 0, C[2] -> 0})

Plot[-slt, {x, -1 , 1}, AxesLabel -> {"x", "y(x)"}, PerformanceGoal -> "Speed"]

-slt[[2]] /. x -> 1