Seems to be very difficult. At least you can get an impression of the solution by a numerical method....

Attachments:

It appears that Mathematica cannnot solve it symbolically.

In[5]:= DSolve[{y''''[x] + 12/x y'''[x] + 36/x^2 y''[x] + 
    24/x^3 y'[x] + 60 (7 - 18 x^4 + 3 x^8) y[x]^9 == 0, y[0] == 1, 
  y'[0] == 0, y''[0] == 0, y'''[0] == 0}, y[x], x]

Out[5]= DSolve[{60 (7 - 18 x^4 + 3 x^8) y[x]^9 + (
    24 Derivative[1][y][x])/x^3 + (36 (y^\[Prime]\[Prime])[x])/x^2 + (
    12 
\!\(\*SuperscriptBox[\(y\), 
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction->None]\)[x])/x + 
\!\(\*SuperscriptBox[\(y\), 
TagBox[
RowBox[{"(", "4", ")"}],
Derivative],
MultilineFunction->None]\)[x] == 0, y[0] == 1, 
  Derivative[1][y][0] == 0, (y^\[Prime]\[Prime])[0] == 0, 
\!\(\*SuperscriptBox[\(y\), 
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction->None]\)[0] == 0}, y[x], x]

Easier to diagnose problems when they are posted in code sample windows (first icon). However, I'm guessing you've got uncleared expressions that are resulting in the True that you see.

I write argument of y'' but now the error is "

DSolve::deqn: Equation or list of equations expected instead of True
in the first argument {60 x^3 (7-18 x^2+3 x^8) y[x]^9+24 (y^\[Prime])[x]
+36 x (y^\[Prime]\[Prime])[x]+12 x^2 (y^(3))[x]+x^3 (y^(4))[x]==0,True,0,True,True}. >>