A first step might be to solve the homogeneous equation:
In[13]:= DSolve[D[y[x], {x, 4}] - x y[x] == 0, y[x], x]
Out[13]= {{y[x] ->
C[1] HypergeometricPFQ[{}, {2/5, 3/5, 4/5}, x^5/625] + (1/(5^(
4/5)))(-1)^(1/5)
x C[2] HypergeometricPFQ[{}, {3/5, 4/5, 6/5}, x^5/625] + (1/(
5 5^(3/5)))(-1)^(2/5)
x^2 C[3] HypergeometricPFQ[{}, {4/5, 6/5, 7/5}, x^5/625] + (1/(
25 5^(2/5)))(-1)^(3/5)
x^3 C[4] HypergeometricPFQ[{}, {6/5, 7/5, 8/5}, x^5/625]}}
then try one of the standard techniques for the corresponding inhomogeneous equation.
For a numerical solution
In[19]:= NDSolve[{D[y[x], {x, 4}] -
x y[x] == -(11 + 9 x + x^2 - x^3) E^x, y[-1] == 0, y[1] == 0,
y'[-1] == 2/E, y'[1] == -2/E}, y[x], x]
Out[19]= {{y[x] -> \!\(\*
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