I removed the constants to make the ODE simple and used constants for initial conditions, which can be modified, accordingly.

(* 1st approach *)
dsolSDL03 = DSolve[
  {D[y[x], {x, 4}] + D[y[x], {x, 2}] == 0,
   y[0] == 4, y'[0] == 3, y''[0] == 2, y'''[0] == 1 },
  {y}, {x, 0, 10}]

(* 2nd approach *)
dsolSDL04 = DSolve[
  {D[y0[x], {x, 1}] == y1[x]    (* 1st derivative of y[x] *),
   D[y1[x], {x, 1}] == y2[x]    (* 2nd derivative of y[x] *),
   D[y2[x], {x, 1}] == y3[x]    (* 3rd derivative of y[x] *),
   D[y3[x], {x, 1}] == -y2[x]   (* 4th derivative of y[x] *),
   y0[0] == 4, y1[0] == 3, y2[0] == 2, y3[0] == 1 },
  {y0, y1, y2, y3}, {x, 0, 10}]

{Plot[Evaluate[{y[x], y'[x], y''[x], y'''[x]} /. dsolSDL03], {x, 0, 10}],
 Plot[Evaluate[{y0[x], y1[x], y2[x], y3[x]} /. dsolSDL04], {x, 0, 10}]}