\[ (x-2) (x-1) y''(x)-(2 x-3) y'(x)+y(x)=0 \] ✓ Mathematica : cpu = 0.0322662 (sec), leaf count = 64
DSolve[y[x] - (-3 + 2*x)*Derivative[1][y][x] + (-2 + x)*(-1 + x)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (x^2-3 x+2\right ) P_{\frac {1}{2} \left (-1+\sqrt {5}\right )}^2(2 x-3)+c_2 \left (x^2-3 x+2\right ) Q_{\frac {1}{2} \left (-1+\sqrt {5}\right )}^2(2 x-3)\right \}\right \}\] ✓ Maple : cpu = 0.609 (sec), leaf count = 93
dsolve((x-1)*(x-2)*diff(diff(y(x),x),x)-(2*x-3)*diff(y(x),x)+y(x)=0,y(x))
\[y \left (x \right ) = \left (x -2\right )^{2} \left (c_{2} \hypergeom \left (\left [\frac {5}{2}+\frac {\sqrt {5}}{2}, \frac {1}{2}+\frac {\sqrt {5}}{2}\right ], \left [\sqrt {5}+1\right ], \frac {1}{x -1}\right ) \left (x -1\right )^{-\frac {1}{2}-\frac {\sqrt {5}}{2}}+c_{1} \hypergeom \left (\left [\frac {5}{2}-\frac {\sqrt {5}}{2}, \frac {1}{2}-\frac {\sqrt {5}}{2}\right ], \left [-\sqrt {5}+1\right ], \frac {1}{x -1}\right ) \left (x -1\right )^{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\]