Let's try the simple thing last

In[208]:= Clear[aliEq]
aliEq[f_, x_] := (b + a x^2)/x^3 D[f, x] + (a - b/x) D[f, {x, 2}] + 2 b/x D[f, {x, 3}] + b D[f, {x, 4}] /; ! FreeQ[f, x]

In[210]:= Clear[aq]
aq = aliEq[Sum[a[o] x^o, {o, 1, n}], x];

In[212]:= Coefficient[aq /. n -> 100, x, #] & /@ Range[20] 
Out[212]= {9 a a[3] - 12 b a[4] + 245 b a[5], 
          16 a a[4] - 20 b a[5] + 606 b a[6], 
          25 a a[5] - 30 b a[6] + 1267 b a[7], 
          36 a a[6] - 42 b a[7] + 2360 b a[8], 
          49 a a[7] - 56 b a[8] + 4041 b a[9], 
          64 a a[8] - 72 b a[9] + 6490 b a[10], 
          81 a a[9] - 90 b a[10] + 9911 b a[11], 
         100 a a[10] - 110 b a[11] + 14532 b a[12], 
         121 a a[11] - 132 b a[12] + 20605 b a[13], 
         144 a a[12] - 156 b a[13] + 28406 b a[14], 
         169 a a[13] - 182 b a[14] + 38235 b a[15], 
         196 a a[14] - 210 b a[15] + 50416 b a[16], 
         225 a a[15] - 240 b a[16] + 65297 b a[17], 
         256 a a[16] - 272 b a[17] + 83250 b a[18], 
         289 a a[17] - 306 b a[18] + 104671 b a[19], 
         324 a a[18] - 342 b a[19] + 129980 b a[20], 
         361 a a[19] - 380 b a[20] + 159621 b a[21], 
         400 a a[20] - 420 b a[21] + 194062 b a[22], 
         441 a a[21] - 462 b a[22] + 233795 b a[23], 
         484 a a[22] - 506 b a[23] + 279336 b a[24]}

In[217]:= FindSequenceFunction[{245, 606, 1267, 2360, 4041, 6490, 9911, 14532, 20605, 28406, 38235, 50416}]
Out[217]= 76 + 103 #1 + 53 #1^2 + 12 #1^3 + #1^4 &

In[228]:= Clear[coeff]
coeff[n_] := (n + 2)^2 a a[n + 2] - (n + 2) (n + 3) b a[n + 3] + (76 + 103 n + 53 n^2 + 12 n^3 + n^4) b a[n + 4];

In[230]:= coeff[20]
Out[230]= 484 a a[22] - 506 b a[23] + 279336 b a[24]

the job is to solve all this equations to 0 in the homogenous case or to be equal to the corresponding coefficient of the inhomogenity p[r]. The system seems not to decouple, so the differential equation remains unsolved.

thank u

a,b are constant.

and my boundary conditions are:

w[d]==0 ,

w[0]= = limit (cant be infinite i.e if there is any "r" in denominator then the Corresponding Factor must be zero for example if we have a term like this: " C[1] /r + a*r^2 " then C[1] must be zero for the sake of not being infinite ,

D[D[w[r], r], r] + 1/r D[w[r], r]== limit ,

D[b (D[D[w[r], r], r] + 1/r D[w[r], r]), r]== limit

how can i solve it with this boundary conditions?

one of the boundary condition is (D[D[w[r], r], r] + 1/r D[w[r], r]) when r=0 ,cant be infinite

Still we don't know what the existence interval for the independent variable r is. If we know it, recognizing such a condition (finiteness condition, being zero at inifinity etc.) is a matter of a so called ansatz (a.k.a. educated guess).

Possibly you know the most famous example: Hermann Weyl helped Erwin Schrödinger to solve the Schrödinger equation for the hydrogen atom.

That's an ordinary differential equation of fourth degree for w[r] with inhomogenity p[r]. To make this work you need to specify

• an existence interval for the independent variable r
• four boundary conditions for w, w', w'', w''', see OrdinaryDifferentialEquation
• the inhomogenity p[r] as a function
• a and b

Then most probably you have to use NDSolve because of p[r].

If you're a lover of step-wise thinking try to solve the homogenous equation (p[r] === 0) first.

Be aware of the singularity if r = 0 belongs to the existence interval.