Clear["Global`*"]

  
order = 
        0;
Null

\( (*\ Eq . \ 2.7  .2\ *) 

    P[x_] := ∑\+\(n = 0\)\%order p[
            n]\ \((x - x0)\)\^\(\(n\)\(\ \ \)\)\)
Print["P[x]=", P[x]]

Q[
        x_] := (∑_(n = 0)^order) q[
            n] (((x - x0))^(n  ))
Print["Q[x]=", Q[x]]

\( (*\ Eq . \ 2.7  .5\ *) 

    U[x_] := \(\(\((x - 
                x0)\)\^α\)\(\ \ \)\((1 + ∑\+\(n = 1\)\%order u[
                  n]\ \((x - x0)\)\^n)\)\(\ \)\)\)
Print["U[x]=", U[x]]

\( (*\ 
      Differential\ Equation\ 2.7  .1\ *) 
\(\(eq = \((x - x0)\)\^2\ \(U''\)[
              x] + \((x - x0)\)\ P[x]\ \(U'\)[x] + Q[x]\ U[x] == 
        0\)\(
\)
    \)\)
sol1 = Flatten[Solve[eq, α]]

order = 
        1;
P[x_] := (∑_(n = 0)^order) p[
            n] (((x - x0))^(n  ))
Q[x_] := (∑_(n = 0)^order) q[
            n] (((x - x0))^(n  ))
U[x_] := (((x - 
                x0))^α)  (1 + (∑_(n = 1)^order) u[
                  n] (((x - x0))^n)) 
eq = (((x - x0))^2) U''[x] + (x - x0) P[x] U'[x] + 
          Q[x] U[x] == 0
FullSimplify[
      Solve[eq /. sol1[[1]], u[1]]]
FullSimplify[Solve[eq /. sol1[[2]], u[1]]]