NDSolve[{y''[
    x] == ((Sqrt[(Integrate[y[x], x])^2 + (Integrate[x[y],
             y])^2]*0.5*((1.2276506087738044` -
           0.058867354927993866` y[x] +
           0.013674072942291457` y[x]^2 -
           0.006683664467058859` y[x]^3 +
           0.0012088043510569788` y[x]^4 -
           0.00012105199792680971` y[x]^5 +
           7.777141557633639`*^-6 y[x]^6 -
           3.4251038818338157`*^-7 y[x]^7 +
           1.0698730986217082`*^-8 y[x]^8 -
           2.409706752959389`*^-10 y[x]^9 +
           3.927274987489023`*^-12 y[x]^10 -
           4.5886287468034394`*^-14 y[x]^11 +
           3.746561723569994`*^-16 y[x]^12 -
           2.0294454009615562`*^-18 y[x]^13 +
           6.551643589092214`*^-21 y[x]^14 -
           9.539267389433688`*^-24 y[x]^15))*0.292247)/
      817)*(Integrate[x[y], y]*1.71*a -
      Integrate[y[x], x]*(1.84*a^2 + 0.27*a + 0.017)),
  x''[y] == ((Sqrt[(Integrate[y[x], x])^2 + (Integrate[x[y],
              y])^2]*0.5*((1.2276506087738044` -
            0.058867354927993866` y[x] +
            0.013674072942291457` y[x]^2 -
            0.006683664467058859` y[x]^3 +
            0.0012088043510569788` y[x]^4 -
            0.00012105199792680971` y[x]^5 +
            7.777141557633639`*^-6 y[x]^6 -
            3.4251038818338157`*^-7 y[x]^7 +
            1.0698730986217082`*^-8 y[x]^8 -
            2.409706752959389`*^-10 y[x]^9 +
            3.927274987489023`*^-12 y[x]^10 -
            4.5886287468034394`*^-14 y[x]^11 +
            3.746561723569994`*^-16 y[x]^12 -
            2.0294454009615562`*^-18 y[x]^13 +
            6.551643589092214`*^-21 y[x]^14 -
            9.539267389433688`*^-24 y[x]^15))*0.292247*1.71*a)/
       817)*(Integrate[y[x], x]*1.71*a -
       Integrate[x[y],
         y]*(1.84*a^2 + 0.27*a +
          0.017)) - (-(398600.4418*10^(9)*817)/((y[x] +
           6371222)^2))*9.80665, x[0] == 1, y[0] == 160000}, {y[x],
  x[y]}, {x, y}]