ODE
\[ -2 a x y'(x)+a (a+1) y(x)+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0216746 (sec), leaf count = 105
\[\left \{\left \{y(x)\to x^{\frac {-\sqrt {\frac {1}{a^2+a}} \sqrt {a}-\sqrt {\frac {1}{a^2+a}} a^{3/2}+2 \sqrt {a+1} a+\sqrt {a+1}}{2 \sqrt {a+1}}} \left (c_2 x^{\sqrt {a} \sqrt {a+1} \sqrt {\frac {1}{a^2+a}}}+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.01 (sec), leaf count = 17
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{1+a}+{\it \_C2}\,{x}^{a} \right \} \] Mathematica raw input
DSolve[a*(1 + a)*y[x] - 2*a*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x^((Sqrt[1 + a] + 2*a*Sqrt[1 + a] - Sqrt[a]*Sqrt[(a + a^2)^(-1)] - a^(3/2)*Sqrt[(a + a^2)^(-1)])/(2*Sqrt[1 + a]))*(C[1] + x^(Sqrt[a]*Sqrt[1 + a]*Sqrt[(a + a^2)^(-1)])*C[2])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)-2*a*x*diff(y(x),x)+a*(1+a)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*x^(1+a)+_C2*x^a