Extract coefficients from ODE

Question

I'm trying to extract the coefficients from an ODE $$y''''+y''+x^2 y'-6y+ 8\ cos(x)$$
and hoping the code will give a list with all the coefficients in, ranging from lower order to higher order, i.e. $$\{8\ cos(x)\,,\,-6y\,,\,x^2\,,\, 1\,,\,1\}$$

So far I can only extract the $x^2$ and $-6$ in $x^2 y'$ and $-6y$ using

In[1] : Cases[ y'''' + y'' + x^2 y' - 6 y + 8 Cos[x],  
        Times[x_, Derivative[_][_]] :> x, Infinity]  
Out[2]: {x^2}  

In[3] : Cases[ y'''' + y' + x^2 y' - 6 y + 8 Cos[x], x__ y :> x, Infinity]  
Out[4]: {-6}

How can I deal with the $y'''$ and $y''$ which do not have the Times header before them? I've tried this one which just return me all the $y^{(n)}$s.

In[5]: Cases[ y'''' + y'' + x^2 y' - 6 y + 8 Cos[x], 
       Derivative[_][y], Infinity]  
Out[6]: {y',y'',y''''}  

Answer 1

Given that it's linear, I would probably do it this way:

CoefficientList[
 y'''' + y'' + x^2 y' - 6 y + 8 Cos[x] /. 
  Derivative[n_][y] :> y^(n + 1), {y}]
(*  {8 Cos[x], -6, x^2, 1, 0, 1}  *)

Note that the output here has the coefficient of y''', which the desired output in the OP omits. I see every reason to include it, though.

---

More straightforward:

Flatten@CoefficientArrays[y'''' + y'' + x^2 y' - 6 y + 8 Cos[x], 
  Table[Derivative[n][y], {n, 0, 4}]]
(*  {8 Cos[x], -6, x^2, 1, 0, 1}  *)