\[ y''(x)=\frac {(x-4) y'(x)}{2 (x-2) x}-\frac {(x-3) y(x)}{2 (x-2) x^2} \] ✓ Mathematica : cpu = 0.0314324 (sec), leaf count = 55
DSolve[Derivative[2][y][x] == -1/2*((-3 + x)*y[x])/((-2 + x)*x^2) + ((-4 + x)*Derivative[1][y][x])/(2*(-2 + x)*x),y[x],x]
\[\left \{\left \{y(x)\to \frac {c_1 \sqrt [4]{x-2} \sqrt {x}}{\sqrt [4]{2-x}}+\frac {2 c_2 (x-2)^{3/4} \sqrt {x}}{\sqrt [4]{2-x}}\right \}\right \}\] ✓ Maple : cpu = 0.02 (sec), leaf count = 19
dsolve(diff(diff(y(x),x),x) = 1/2/x*(x-4)/(x-2)*diff(y(x),x)-1/2*(x-3)/x^2/(x-2)*y(x),y(x))
\[y \left (x \right ) = c_{1} \sqrt {x}+c_{2} \sqrt {x \left (x -2\right )}\]