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 = 0.4109 (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 = 0.596 (sec), leaf count = 156
\[\left [y \left (x \right ) = \textit {\_C1} \left (x^{2}+1\right )^{1+\frac {a}{2}-\frac {b}{2}} \hypergeom \left (\left [\frac {3}{4}+\frac {a}{2}-\frac {b}{4}+\frac {\sqrt {b^{2}-2 b -4 c +1}}{4}, \frac {3}{4}+\frac {a}{2}-\frac {b}{4}-\frac {\sqrt {b^{2}-2 b -4 c +1}}{4}\right ], \left [\frac {a}{2}+\frac {1}{2}\right ], -x^{2}\right )+\textit {\_C2} \,x^{1-a} \left (x^{2}+1\right )^{1+\frac {a}{2}-\frac {b}{2}} \hypergeom \left (\left [\frac {5}{4}-\frac {b}{4}+\frac {\sqrt {b^{2}-2 b -4 c +1}}{4}, \frac {5}{4}-\frac {b}{4}-\frac {\sqrt {b^{2}-2 b -4 c +1}}{4}\right ], \left [-\frac {a}{2}+\frac {3}{2}\right ], -x^{2}\right )\right ]\] 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))
Maple raw output
[y(x) = _C1*(x^2+1)^(1+1/2*a-1/2*b)*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*a+1/2],-x^2)+_C2*x^(1-a)*(x^2+1)^(1+1/2*a-1/2*b)*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)]