This is not an exact equation

In[160]:= Clear[a0, a1, a2, a3, a4]
a0[r_] := b
a1[r_] := 2 b/r
a2[r_] := (a - b/r)
a3[r_] := (b + a r^2)/r^3
a4[r_] := 0

In[167]:= (-1)^4 D[a0[r], {r, 4}] + (-1)^3 D[a1[r], {r, 3}] + (-1)^2 D[a2[r], {r, 2}] + (-1)^1 D[a3[r], {r, 1}] + a4[r] // Simplify
Out[167]= (15 b - 2 b r + a r^2)/r^4

unless a and b are both zero. So one cannot reduce the order of the equation. Nevertheless - in experimental mode - set v[r] = r^3 w'[r] to arrive at

(* Ansatz: w'[r] = r^3 v[r] *)
(b + a r^2) v[r] + (a - b/r) D[r^3 v[r], r] + 
  2 b/r D[r^3 v[r], {r, 2}] + b D[r^3 v[r], {r, 3}] // Simplify
(b (19 - 3 r) + 4 a r^2) v[r] + r ((30 b - b r + a r^2) v'[r] + b r (11 (v''[r] + r v'''[r]))

for this new equation it is possible to get some inhomogenous solutions (for simple inhomogenities) but it seems impossible to find one which is finite at r = 0.

(* inhomogen, transformiert *)
Clear[s, a, b, v, w, r]
s = With[{d = 2, \[Mu] = 1, \[Nu] = 1},
  DSolve[{(b (19 - 3 r) + 4 a r^2) v[r] + 
      r (30 b - b r + a r^2) v'[r] + 11 b r^2  v''[r] + 
      b r^3 v'''[r] == r^\[Mu] (d - r)^\[Nu], v[d] == 0, v'[d] == 2, 
    v''[d] == 1}, v, r]]

Clear[f]
f = s[[1, 1, 2]] /. {a -> 1, b -> 2}

Plot[f[x], {x, -1, 5}]

Things which did not work

• prescribing a finiteness condition at r = 0
• get solutions to the inhomogenous equation in less than an hour (computation aborted) with NDSolve
• get solutions to the original equation for simple inhomogenities with DSolve

The homogeneous equation can be solved to give a DifferentialRoot object, to plot something one has to prescribe the values mentioned - done by chance here -

In[13]:= Clear[a, b, r]
DSolve[{(b + a r^2) w'[r] + (a r - b)  r^2 w''[r] + 2 b r^2 w'''[r] + b r^3 w''''[r] == 0, 
         w[1] == 1, w'[1] == 0, w''[1] == 1, w'''[1] == 2}, w, r]

then make it into a function which can be plotted

In[50]:= Clear[f]
f = (Out[14][[1, 1, 2]]) /. {a -> 2, b -> 3}

and plot it

the behaviour depends strongly on a and b. Experiment on your own.

Another thing to do (a.k.a. jobs for the weekend): Specify a and b in an attempt to reach the form of some known (named) differential equation. Check the results. Construct solutions from known solutions.