ODE
\[ (1-x) x y''(x)+2 y'(x)+y(x)=0 \] ODE Classification
[_Jacobi]
Book solution method
TO DO
Mathematica ✓
cpu = 0.146186 (sec), leaf count = 73
\[\left \{\left \{y(x)\to c_2 G_{2,2}^{2,0}\left (x\left |\begin {array}{c} \frac {1}{2} \left (3-\sqrt {5}\right ),\frac {1}{2} \left (3+\sqrt {5}\right ) \\ -1,0 \\\end {array}\right .\right )+c_1 \, _2F_1\left (\frac {1}{2} \left (-1-\sqrt {5}\right ),\frac {1}{2} \left (-1+\sqrt {5}\right );2;x\right )\right \}\right \}\]
Maple ✓
cpu = 0.117 (sec), leaf count = 85
\[ \left \{ y \left ( x \right ) = \left ( -1+x \right ) ^{3} \left ( {\it \_C1}\,{x}^{{\frac {\sqrt {5}}{2}}-{\frac {5}{2}}}{\mbox {$_2$F$_1$}({\frac {3}{2}}-{\frac {\sqrt {5}}{2}},{\frac {5}{2}}-{\frac {\sqrt {5}}{2}};\,-\sqrt {5}+1;\,{x}^{-1})}+{\it \_C2}\,{x}^{-{\frac {5}{2}}-{\frac {\sqrt {5}}{2}}}{\mbox {$_2$F$_1$}({\frac {3}{2}}+{\frac {\sqrt {5}}{2}},{\frac {5}{2}}+{\frac {\sqrt {5}}{2}};\,\sqrt {5}+1;\,{x}^{-1})} \right ) \right \} \] Mathematica raw input
DSolve[y[x] + 2*y'[x] + (1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric2F1[(-1 - Sqrt[5])/2, (-1 + Sqrt[5])/2, 2, x] + C[2]*MeijerG[{{}, {(3 - Sqrt[5])/2, (3 + Sqrt[5])/2}}, {{-1, 0}, {}}, x]}}
Maple raw input
dsolve(x*(1-x)*diff(diff(y(x),x),x)+2*diff(y(x),x)+y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (-1+x)^3*(_C1*x^(1/2*5^(1/2)-5/2)*hypergeom([3/2-1/2*5^(1/2), 5/2-1/2*5^(1/2)],[-5^(1/2)+1],1/x)+_C2*x^(-5/2-1/2*5^(1/2))*hypergeom([3/2+1/2*5^(1/2), 5/2+1/2*5^(1/2)],[5^(1/2)+1],1/x))