\[ a y'(x)+(x-1) x y''(x)-2 y(x)=0 \] ✓ Mathematica : cpu = 0.622205 (sec), leaf count = 360
DSolve[-2*y[x] + a*Derivative[1][y][x] + (-1 + x)*x*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {c_2 x^a \left (a^2+2 a x-a+2 x^2-2 x\right ) (1-x)^{-a} \left (-\frac {a \, _2F_1\left (1,-a;1-a;\frac {\left (-a+\sqrt {1-a^2}+1\right ) (x-1)}{\left (-a+\sqrt {1-a^2}-1\right ) x}\right )}{\left (1-a^2\right )^{3/2}}+\frac {a \, _2F_1\left (1,-a;1-a;\frac {\left (a+\sqrt {1-a^2}-1\right ) (x-1)}{\left (a+\sqrt {1-a^2}+1\right ) x}\right )}{\left (1-a^2\right )^{3/2}}+\frac {(x-1) \left (\left (\sqrt {1-a^2}+1\right ) \, _2F_1\left (2,1-a;2-a;\frac {\left (-a+\sqrt {1-a^2}+1\right ) (x-1)}{\left (-a+\sqrt {1-a^2}-1\right ) x}\right )-\left (\sqrt {1-a^2}-1\right ) \, _2F_1\left (2,1-a;2-a;\frac {\left (a+\sqrt {1-a^2}-1\right ) (x-1)}{\left (a+\sqrt {1-a^2}+1\right ) x}\right )\right )}{\left (a^2-1\right )^2 x}\right )}{2 a^2 \left (a^2+3 a+4\right )}+\frac {c_1 \left (a^2+2 a x-a+2 x^2-2 x\right )}{a^2+3 a+4}\right \}\right \}\] ✓ Maple : cpu = 0.016 (sec), leaf count = 42
dsolve(x*(x-1)*diff(diff(y(x),x),x)+a*diff(y(x),x)-2*y(x)=0,y(x))
\[y \left (x \right ) = \left (a^{2}+a \left (2 x -1\right )+2 x^{2}-2 x \right ) c_{1}+c_{2} \left (x -1\right )^{-a} \left (x -1\right ) x^{a} x\]