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The following code solves a simple 2nd-order linear differential equation with all real parameters.

FullSimplify[
 DSolve[(4 q^2 r - b^2 r^3 + 8 p q s + (4 p^2 s^2)/r - 4 r m^2)/(4 r)
      f[y] - b^2 r^2 y (2 D[f[y], y] + 
       y D[f[y], {y, 2}]) == 0, f[y], y, 
  Assumptions -> Element[q | r | p | s | b | m, Reals]], 
 Assumptions -> Element[q | r | p | s | b | m, Reals]]

The result is as horrible as$$y^{\frac{1}{2} \left(-\frac{\sqrt{r^2 \left(b^2 r^2+4 m^2-4 q^2\right)-4 p^2 s^2-8 p q r s} \sqrt{\frac{r^4}{r^2 \left(b^2 r^2+4 m^2-4 q^2\right)-4 p^2 s^2-8 p q r s}-\frac{1}{b^2}}}{r^2}-1\right)} \left(c_2 y^{\frac{\sqrt{r^2 \left(b^2 r^2+4 m^2-4 q^2\right)-4 p^2 s^2-8 p q r s} \sqrt{\frac{r^4}{r^2 \left(b^2 r^2+4 m^2-4 q^2\right)-4 p^2 s^2-8 p q r s}-\frac{1}{b^2}}}{r^2}}+c_1\right).$$ However, without any further assumption, I found by hand that the solution is actually just $$c_2 y^{\frac{\sqrt{D }}{r^2 \left| b\right| }-\frac{1}{2}}+c_1 y^{-\frac{\sqrt{D }}{r^2 \left| b\right| }-\frac{1}{2}},$$ where $D=(ps+qr)^2-r^2m^2$.

I tried various assumptions but no available except directly using $D$ (DD in the code) to replace things from the very begining

Assuming[Element[DD | r | b, Reals], 
f[y] /. DSolve[(DD/r^2 - b^2 r^2/4) f[y] - b^2 r^2 y (2 D[f[y], y] 
+ y D[f[y], {y, 2}]) == 0, f[y], y][[1]] // FullSimplify] // Expand

The result is exactly what I found by hand. But this is no more than Monday morning quarterback. Note that mathematically there should be no difference between assuming Element[DD | r | b, Reals] and Element[q | r | p | s | b | m, Reals].

So how can one obtain this in a somewhat elegant or at least general manner? I still hope some approach 'better' than or different from @Bob Hanlon 's answer.

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Clear["Global`*"]

eqn = (4 q^2 r - b^2 r^3 + 8 p q s + (4 p^2 s^2)/r - 4 r m^2)/(4 r) 
    f[y] - b^2 r^2 y (2 D[f[y], y] + y D[f[y], {y, 2}]) == 0;

expr1 = Assuming[Element[q | r | p | s | b | m, Reals],
   f[y] /. DSolve[eqn, f[y], y][[1]] // FullSimplify] // Expand

(* y^(1/2 (-1 - (
     Sqrt[r^2 (4 m^2 - 4 q^2 + b^2 r^2) - 8 p q r s - 4 p^2 s^2]
       Sqrt[-(1/b^2) + r^4/(
       r^2 (4 m^2 - 4 q^2 + b^2 r^2) - 8 p q r s - 4 p^2 s^2)])/r^2)) C[1] + 
 y^((Sqrt[r^2 (4 m^2 - 4 q^2 + b^2 r^2) - 8 p q r s - 4 p^2 s^2]
     Sqrt[-(1/b^2) + r^4/(
     r^2 (4 m^2 - 4 q^2 + b^2 r^2) - 8 p q r s - 4 p^2 s^2)])/r^2 + 
   1/2 (-1 - (
      Sqrt[r^2 (4 m^2 - 4 q^2 + b^2 r^2) - 8 p q r s - 4 p^2 s^2]
        Sqrt[-(1/b^2) + r^4/(
        r^2 (4 m^2 - 4 q^2 + b^2 r^2) - 8 p q r s - 4 p^2 s^2)])/r^2)) C[2] *)

Assuming temporarily that the conditions are met such that Sqrt[a] Sqrt[b] == Sqrt[a*b]

expr2 = expr1 /. Sqrt[a_] Sqrt[b_] -> Sqrt[a*b] // Simplify // Expand

(* y^(-(1/2) - Sqrt[(-m^2 r^2 + (q r + p s)^2)/b^2]/r^2) C[1] + 
 y^(-(1/2) + Sqrt[(-m^2 r^2 + (q r + p s)^2)/b^2]/r^2) C[2] *)

Showing that the expr2 satisfies eqn more generally,

eqn /. {f -> Function[{y}, Evaluate@expr2]} // Simplify

(* True *)
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  • $\begingroup$ Thanks. I know what you mean. But it actually doesn't falsify my simplification. As I mentioned in the question, you can easily see this by using $\Delta$ from the very beginning. Assuming[Element[DD | r | b, Reals], f[y] /. DSolve[(DD/r^2 - b^2 r^2/4) f[y] - b^2 r^2 y (2 D[f[y], y] + y D[f[y], {y, 2}]) == 0, f[y], y][[1]] // FullSimplify] // Expand They are mathematically the same thing. $\endgroup$ – xiaohuamao Jan 5 at 7:57
  • $\begingroup$ It's easy to see for the invalid case you claimed, the two solutions simply interchange and the whole expression stays the same. So your answer doesn't really answer it. $\endgroup$ – xiaohuamao Jan 5 at 17:22
  • $\begingroup$ Thank you for the edit. It looks as though MMA 'oversimplifies' a little, albeit correctly, rendering it not easy to recover what one would normally want without ad hoc assumptions... $\endgroup$ – xiaohuamao Jan 6 at 6:00

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