As the manual says

Apply N[DSolve[...]] to invoke NDSolve if symbolic solution fails:

In[13]:= With[{a = 2},
 DSolve[{y''[x] == y[x]/(x^2 + a^2)^(3/2), y'[0] == 1, y[0] == 2}, y, {x, 0, 4}]
 ]
Out[13]= DSolve[{(y^\[Prime]\[Prime])[x] == y[x]/(4 + x^2)^(3/2), Derivative[1][y][0] == 1, y[0] == 2}, y, {x, 0, 4}]

In[14]:= N[%]
Out[14]= {{y -> InterpolatingFunction[{{0., 4.}}, <>]}}

Numerical solution urges you to define boundary conditions and an evaluation interval for the independent variable x. For some integer exponents

DSolve[y''[x] == y[x]/(x^2 + a^2), y, x]
DSolve[y''[x] == y[x]/(x^2 + a^2)^2, y, x]

DSolve does not return the input.