Some information on Pietro's ODE:
$$
(4x^3-136x^2+4x)u''+(8x^2-204x+4)u'+(x-10)u=0
\tag{1}$$
I will use this notation:
$$
c := (1+\sqrt{2}\;)^4 = 17+12\sqrt{2} \approx 33.97056 ,
\\
c_o := \frac{1}{c} = 34-c = 17-12\sqrt{2} \approx 0.02944 ,
\\
a := 1-c^2 = -576-408\sqrt{2} \approx -1159.9991 ,
\\
q := -\frac{11317}{4}-234\sqrt{2} \approx -660.176 ,
\\
\alpha := \frac{3}{2}, \beta := \frac{3}{2}, \gamma := \frac{3}{2}, 
\delta := 1, \epsilon := \frac{3}{2} .
$$
Maple converts $(1)$ to a Heun differential equation, and evaluates it in terms of the Heun functions.  See [DLMF][1] for information on that.  I will follow their notation.  In interval $(c_o,c)$, two linearly independent solutions of $(1)$ are
$$
u_1(x) := (x-c_o)^{1/2}(c-x)^{1/2} Hl\big(a,q;\alpha,\beta,\gamma,\delta;1-cx\big)
\\
= \sqrt {-{x}^{2}+34\,x-1}\;{\it Hl} \left( -408\,\sqrt {2}-576,-234\,
\sqrt {2}-{\frac {1317}{4}};\frac{3}{2},\frac{3}{2},\frac{3}{2},1,-17\;x-12\,x\sqrt {2}+1 \right) 
\\
u_2(x) := (c-x)^{1/2} Hl\big(a,(a\delta+\epsilon)(1-\gamma)+q;\alpha+1-\gamma,\beta+1-\gamma,2-\gamma,\delta;1-cx\big)
\\
= \sqrt {-x+17+12\,\sqrt {2}}\;
{\it Hl} \left( -408\,\sqrt {2}-576,-42-30\,\sqrt {2},1,1,\frac{1}{2},1,-17
\,x-12\,x\sqrt {2}+1 \right) 
$$
The endpoints $x=c_o$ and $x=c$ correspond to $1-cx$ at the Heun singularities $0$ and $a$, respectively.

Here is a graph of $u_1(x)$ 

![u1 all][2]

At the left end, $x=c_o$, it goes to zero, but has a vertical tangent (like a square-root)  

![u1 left][3]

At the right end, $x=c$, it goes to a finite limit, but has a vertical tangent

![u1 right][4]

The wiggles are not real, but show Maple's difficulty in evaluating close to the singularity.

Here is a graph of $u_2(x)$  

![u2 all][5]

At the left end $x=c_o$ it approaches a definite value $2^{7/4}3^{1/2}$ with a non-vertical tangent  

![u2 left][6]  

and at the right end $x=c$ it approaches a definite value with vertical tangent

![u2 right][7]

For some linear combinations of $u_1$ and $u_2$ the square-root term cancels and it approaches a definite limit at $x=c$ with non-zero tangent.  But for those, the approach at $x=c_o$ has vertical tangent.

  [1]: http://dlmf.nist.gov/31
  [2]: https://i.sstatic.net/x7uXD.jpg
  [3]: https://i.sstatic.net/wD5YF.jpg
  [4]: https://i.sstatic.net/8uY3d.jpg
  [5]: https://i.sstatic.net/BVLW0.jpg
  [6]: https://i.sstatic.net/OjwES.jpg
  [7]: https://i.sstatic.net/vOG3m.jpg