\[ y''(x)=\frac {2 y'(x)}{(x-2) x}-\frac {y(x)}{(x-2) x^2} \] ✓ Mathematica : cpu = 0.126153 (sec), leaf count = 104
DSolve[Derivative[2][y][x] == -(y[x]/((-2 + x)*x^2)) + (2*Derivative[1][y][x])/((-2 + x)*x),y[x],x]
\[\left \{\left \{y(x)\to \left (-\frac {1}{2}\right )^{-\frac {1}{\sqrt {2}}} c_1 x^{-\frac {1}{\sqrt {2}}} \, _2F_1\left (-\frac {1}{\sqrt {2}},-1-\frac {1}{\sqrt {2}};1-\sqrt {2};\frac {x}{2}\right )+\left (-\frac {1}{2}\right )^{\frac {1}{\sqrt {2}}} c_2 x^{\frac {1}{\sqrt {2}}} \, _2F_1\left (\frac {1}{\sqrt {2}},-1+\frac {1}{\sqrt {2}};1+\sqrt {2};\frac {x}{2}\right )\right \}\right \}\] ✓ Maple : cpu = 0.472 (sec), leaf count = 81
dsolve(diff(diff(y(x),x),x) = 2/x/(x-2)*diff(y(x),x)-1/x^2/(x-2)*y(x),y(x))
\[y \left (x \right ) = \left (x -2\right )^{2} \left (c_{1} \hypergeom \left (\left [2-\frac {\sqrt {2}}{2}, 1-\frac {\sqrt {2}}{2}\right ], \left [1-\sqrt {2}\right ], \frac {x}{2}\right ) x^{-\frac {\sqrt {2}}{2}}+c_{2} \hypergeom \left (\left [2+\frac {\sqrt {2}}{2}, 1+\frac {\sqrt {2}}{2}\right ], \left [1+\sqrt {2}\right ], \frac {x}{2}\right ) x^{\frac {\sqrt {2}}{2}}\right )\]