4.30.17 \(a y(x)+x^2 y''(x)-2 (1-x) y'(x)=0\)

ODE
\[ a y(x)+x^2 y''(x)-2 (1-x) y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0484321 (sec), leaf count = 145

\[\left \{\left \{y(x)\to 2^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (\frac {1}{x}\right )^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (2^{\sqrt {1-4 a}} c_2 \left (\frac {1}{x}\right )^{\sqrt {1-4 a}} \, _1F_1\left (\frac {1}{2} \left (\sqrt {1-4 a}+1\right );\sqrt {1-4 a}+1;-\frac {2}{x}\right )+c_1 \, _1F_1\left (\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a};1-\sqrt {1-4 a};-\frac {2}{x}\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.041 (sec), leaf count = 47

\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{x}^{-1}}}\sqrt {{x}^{-1}} \left ( {{\sl K}_{{\frac {1}{2}\sqrt {1-4\,a}}}\left ({x}^{-1}\right )}{\it \_C2}+{{\sl I}_{{\frac {1}{2}\sqrt {1-4\,a}}}\left ({x}^{-1}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input

DSolve[a*y[x] - 2*(1 - x)*y'[x] + x^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> 2^(1/2 - Sqrt[1 - 4*a]/2)*(x^(-1))^(1/2 - Sqrt[1 - 4*a]/2)*(C[1]*Hyper
geometric1F1[1/2 - Sqrt[1 - 4*a]/2, 1 - Sqrt[1 - 4*a], -2/x] + 2^Sqrt[1 - 4*a]*(
x^(-1))^Sqrt[1 - 4*a]*C[2]*Hypergeometric1F1[(1 + Sqrt[1 - 4*a])/2, 1 + Sqrt[1 -
 4*a], -2/x])}}

Maple raw input

dsolve(x^2*diff(diff(y(x),x),x)-2*(1-x)*diff(y(x),x)+a*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = exp(-1/x)*(1/x)^(1/2)*(BesselK(1/2*(1-4*a)^(1/2),1/x)*_C2+BesselI(1/2*(1-
4*a)^(1/2),1/x)*_C1)