ODE
\[ \left (a+b x^2\right ) y'(x)+c x y(x)+x \left (x^2+1\right ) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.314027 (sec), leaf count = 145
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 c+1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 c+1}-1\right );\frac {a+1}{2};-x^2\right )+c_2 x^{1-a} \, _2F_1\left (\frac {1}{4} \left (-2 a+b-\sqrt {b^2-2 b-4 c+1}+1\right ),\frac {1}{4} \left (-2 a+b+\sqrt {b^2-2 b-4 c+1}+1\right );\frac {3-a}{2};-x^2\right )\right \}\right \}\]
Maple ✓
cpu = 0.136 (sec), leaf count = 143
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}+1 \right ) ^{1+{\frac {a}{2}}-{\frac {b}{2}}} \left ( {x}^{1-a}{\mbox {$_2$F$_1$}({\frac {5}{4}}-{\frac {b}{4}}+{\frac {1}{4}\sqrt {{b}^{2}-2\,b-4\,c+1}},{\frac {5}{4}}-{\frac {b}{4}}-{\frac {1}{4}\sqrt {{b}^{2}-2\,b-4\,c+1}};\,-{\frac {a}{2}}+{\frac {3}{2}};\,-{x}^{2})}{\it \_C2}+{\mbox {$_2$F$_1$}({\frac {3}{4}}+{\frac {a}{2}}-{\frac {b}{4}}+{\frac {1}{4}\sqrt {{b}^{2}-2\,b-4\,c+1}},{\frac {3}{4}}+{\frac {a}{2}}-{\frac {b}{4}}-{\frac {1}{4}\sqrt {{b}^{2}-2\,b-4\,c+1}};\,{\frac {1}{2}}+{\frac {a}{2}};\,-{x}^{2})}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[c*x*y[x] + (a + b*x^2)*y'[x] + x*(1 + x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric2F1[(-1 + b - Sqrt[1 - 2*b + b^2 - 4*c])/4, (-1 + b + Sqrt[1 - 2*b + b^2 - 4*c])/4, (1 + a)/2, -x^2] + x^(1 - a)*C[2]*Hypergeometric2F1[(1 - 2*a + b - Sqrt[1 - 2*b + b^2 - 4*c])/4, (1 - 2*a + b + Sqrt[1 - 2*b + b^2 - 4*c])/4, (3 - a)/2, -x^2]}}
Maple raw input
dsolve(x*(x^2+1)*diff(diff(y(x),x),x)+(b*x^2+a)*diff(y(x),x)+c*x*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (x^2+1)^(1+1/2*a-1/2*b)*(x^(1-a)*hypergeom([5/4-1/4*b+1/4*(b^2-2*b-4*c+1)^(1/2), 5/4-1/4*b-1/4*(b^2-2*b-4*c+1)^(1/2)],[-1/2*a+3/2],-x^2)*_C2+hypergeom([3/4+1/2*a-1/4*b+1/4*(b^2-2*b-4*c+1)^(1/2), 3/4+1/2*a-1/4*b-1/4*(b^2-2*b-4*c+1)^(1/2)],[1/2+1/2*a],-x^2)*_C1)