ODE
\[ (a+b x) y'(x)+c y(x)+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0728397 (sec), leaf count = 243
\[\left \{\left \{y(x)\to -i^{-\sqrt {b^2-2 b-4 c+1}+b+1} a^{\frac {1}{2} \left (-\sqrt {b^2-2 b-4 c+1}+b-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {b^2-2 b-4 c+1}+b-1\right )} \left (c_1 \, _1F_1\left (\frac {1}{2} \left (b-\sqrt {b^2-2 b-4 c+1}-1\right );1-\sqrt {b^2-2 b-4 c+1};\frac {a}{x}\right )+c_2 i^{2 \sqrt {b^2-2 b-4 c+1}} a^{\sqrt {b^2-2 b-4 c+1}} \left (\frac {1}{x}\right )^{\sqrt {b^2-2 b-4 c+1}} \, _1F_1\left (\frac {1}{2} \left (b+\sqrt {b^2-2 b-4 c+1}-1\right );\sqrt {b^2-2 b-4 c+1}+1;\frac {a}{x}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.184 (sec), leaf count = 114
\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {1}{2}\sqrt {{b}^{2}-2\,b-4\,c+1}}-{\frac {b}{2}}+{\frac {1}{2}}} \left ( {{\sl U}\left (-{\frac {1}{2}}+{\frac {1}{2}\sqrt {{b}^{2}-2\,b-4\,c+1}}+{\frac {b}{2}},\,1+\sqrt {{b}^{2}-2\,b-4\,c+1},\,{\frac {a}{x}}\right )}{\it \_C2}+{{\sl M}\left (-{\frac {1}{2}}+{\frac {1}{2}\sqrt {{b}^{2}-2\,b-4\,c+1}}+{\frac {b}{2}},\,1+\sqrt {{b}^{2}-2\,b-4\,c+1},\,{\frac {a}{x}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[c*y[x] + (a + b*x)*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(I^(1 + b - Sqrt[1 - 2*b + b^2 - 4*c])*a^((-1 + b - Sqrt[1 - 2*b + b^2 - 4*c])/2)*(x^(-1))^((-1 + b - Sqrt[1 - 2*b + b^2 - 4*c])/2)*(C[1]*Hypergeometric1F1[(-1 + b - Sqrt[1 - 2*b + b^2 - 4*c])/2, 1 - Sqrt[1 - 2*b + b^2 - 4*c], a/x] + I^(2*Sqrt[1 - 2*b + b^2 - 4*c])*a^Sqrt[1 - 2*b + b^2 - 4*c]*(x^(-1))^Sqrt[1 - 2*b + b^2 - 4*c]*C[2]*Hypergeometric1F1[(-1 + b + Sqrt[1 - 2*b + b^2 - 4*c])/2, 1 + Sqrt[1 - 2*b + b^2 - 4*c], a/x]))}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+(b*x+a)*diff(y(x),x)+c*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x^(-1/2*(b^2-2*b-4*c+1)^(1/2)-1/2*b+1/2)*(KummerU(-1/2+1/2*(b^2-2*b-4*c+1)^(1/2)+1/2*b,1+(b^2-2*b-4*c+1)^(1/2),1/x*a)*_C2+KummerM(-1/2+1/2*(b^2-2*b-4*c+1)^(1/2)+1/2*b,1+(b^2-2*b-4*c+1)^(1/2),1/x*a)*_C1)