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I have the following differential equation that I'd like to solve:

\begin{equation} \begin{split} &0 = \left[\left(-216x^2+405x-171\right)\sqrt{-3x^2+6x-2}+216x^3-567x^2+429x-90\right]y'(x)\\ &+ \left[\left(3x^2-6x+2\right)\left(48x^2-63x+10\right)-\sqrt{-3x^2+6x-2}\left(144x^3-399x^2+336x-86\right)\right]y''(x)\\ &+ 2\left(-3x^2+6x-2\right)^\frac{3}{2}\left(x\left(\sqrt{-3x^2+6x-2}+3x-5\right)+2\right)y^{(3)}(x) \end{split} \end{equation}

It is a third order linear ODE to which I am searching for a closed form solution. Here's what I entered in to Mathematica

DSolve[((-216 x^2 + 405 x - 171) Sqrt[-3 x^2 + 6 x - 2] + 216 x^3 - 
      567 x^2 + 429 x - 90) y'[
     x] + (-(-3 x^2 + 6 x - 2) (48 x^2 - 63 x + 10) - 
      Sqrt[-3 x^2 + 6 x - 2] (144 x^3 - 399 x^2 + 336 x - 86)) y''[
     x] + 2 (-3 x^2 + 6 x - 2)^(3/
       2) (x (Sqrt[-3 x^2 + 6 x - 2] + 3 x - 5) + 2) y'''[x] == 0, 
 y[x], x]

which is also what was returned. Is it simply that the program cannot handle the higher order (with variable coefficients; moreover, which is not Cauchy-Euler)?

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    $\begingroup$ MMA thinks harder if you include initial conditions y[0]==y0, y'[0]==yp0, y''[0]==ypp0, but still fails to give a closed-form. It even fails to give solutions when specific values for these initial conditions are plugged in. When I try that same approach using NDSolve, I get complex-valued results that are absolutely huge (10^70 and higher) for both real and imaginary parts. That is one nastly-looking equation! $\endgroup$ – Kevin Ausman Jun 11 '19 at 1:08
  • $\begingroup$ Yes, very ugly! $\endgroup$ – Mmmmmm Jun 12 '19 at 1:05
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You can obtain a series solution. Expansion around point other than zero (x=0 makes it non-real). Expansion around x=1 gives analytical solution.

ode=((-216 x^2+405 x-171) Sqrt[-3 x^2+6 x-2]+216 x^3-567 x^2+429 x-90) 
   y'[x]+(-(-3 x^2+6 x-2) (48 x^2-63 x+10)-Sqrt[-3 x^2+6 x-2] 
  (144 x^3-399 x^2+336 x-86)) y''[x]+2 (-3 x^2+6 x-2)^(3/2) 
  (x (Sqrt[-3 x^2+6 x-2]+3 x-5)+2) y'''[x]==0;

 AsymptoticDSolveValue[ode,y[x],{x,1,6}]

$$ c_2 \left(-\frac{4908503 (x-1)^6}{20480}+\frac{299481 (x-1)^5}{2560}+\frac{2315}{128} (x-1)^4-\frac{143}{8} (x-1)^3+x-1\right)+c_3 \left(\frac{1247879 (x-1)^6}{61440}+\frac{64343 (x-1)^5}{7680}-\frac{1915}{384} (x-1)^4-\frac{3}{8} (x-1)^3+\frac{1}{2} (x-1)^2\right)+c_1 $$

There are 3 constants, since it is 3rd order ODE.

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    $\begingroup$ There's something I don't understand about this. As the order increases, the coefficients of order > 2 seem to go to infinity. None of the examples in the docs do this. $\endgroup$ – Michael E2 Jun 12 '19 at 13:21
  • $\begingroup$ @MichaelE2 I see. But there are terms with plus sign and terms with minus sign there. So may be they all smooth themselves? I do not know. But good observation. $\endgroup$ – Nasser Jun 13 '19 at 7:44

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