ODE
\[ -a y'(x)+(1-x) x y''(x)+2 y(x)=0 \] ODE Classification
[[_2nd_order, _exact, _linear, _homogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.852561 (sec), leaf count = 331
\[\left \{\left \{y(x)\to \frac {\left (a^2+a (2 x-1)+2 (x-1) x\right ) \left (\frac {c_2 x^a (1-x)^{-a} \left (-\frac {a \, _2F_1\left (1,-a;1-a;\frac {\left (-a+\sqrt {1-a^2}+1\right ) (x-1)}{\left (-a+\sqrt {1-a^2}-1\right ) x}\right )}{\left (1-a^2\right )^{3/2}}+\frac {a \, _2F_1\left (1,-a;1-a;\frac {\left (a+\sqrt {1-a^2}-1\right ) (x-1)}{\left (a+\sqrt {1-a^2}+1\right ) x}\right )}{\left (1-a^2\right )^{3/2}}+\frac {(x-1) \left (\left (\sqrt {1-a^2}+1\right ) \, _2F_1\left (2,1-a;2-a;\frac {\left (-a+\sqrt {1-a^2}+1\right ) (x-1)}{\left (-a+\sqrt {1-a^2}-1\right ) x}\right )-\left (\sqrt {1-a^2}-1\right ) \, _2F_1\left (2,1-a;2-a;\frac {\left (a+\sqrt {1-a^2}-1\right ) (x-1)}{\left (a+\sqrt {1-a^2}+1\right ) x}\right )\right )}{\left (a^2-1\right )^2 x}\right )}{a^2}+2 c_1\right )}{2 \left (a^2+3 a+4\right )}\right \}\right \}\]
Maple ✓
cpu = 0.066 (sec), leaf count = 42
\[[y \left (x \right ) = \left (a^{2}+2 a x +2 x^{2}-a -2 x \right ) \textit {\_C1} +\textit {\_C2} \,x^{a} x \left (x -1\right )^{-a} \left (x -1\right )]\] Mathematica raw input
DSolve[2*y[x] - a*y'[x] + (1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((a^2 + 2*(-1 + x)*x + a*(-1 + 2*x))*(2*C[1] + (x^a*C[2]*(-((a*Hypergeometric2F1[1, -a, 1 - a, ((1 - a + Sqrt[1 - a^2])*(-1 + x))/((-1 - a + Sqrt[1 - a^2])*x)])/(1 - a^2)^(3/2)) + (a*Hypergeometric2F1[1, -a, 1 - a, ((-1 + a + Sqrt[1 - a^2])*(-1 + x))/((1 + a + Sqrt[1 - a^2])*x)])/(1 - a^2)^(3/2) + ((-1 + x)*((1 + Sqrt[1 - a^2])*Hypergeometric2F1[2, 1 - a, 2 - a, ((1 - a + Sqrt[1 - a^2])*(-1 + x))/((-1 - a + Sqrt[1 - a^2])*x)] - (-1 + Sqrt[1 - a^2])*Hypergeometric2F1[2, 1 - a, 2 - a, ((-1 + a + Sqrt[1 - a^2])*(-1 + x))/((1 + a + Sqrt[1 - a^2])*x)]))/((-1 + a^2)^2*x)))/(a^2*(1 - x)^a)))/(2*(4 + 3*a + a^2))}}
Maple raw input
dsolve(x*(1-x)*diff(diff(y(x),x),x)-a*diff(y(x),x)+2*y(x) = 0, y(x))
Maple raw output
[y(x) = (a^2+2*a*x+2*x^2-a-2*x)*_C1+_C2*x^a*x/((x-1)^a)*(x-1)]