ODE
\[ \left (a x^2+1\right ) y''(x)+b x y'(x)+c y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0774382 (sec), leaf count = 135
\[\left \{\left \{y(x)\to \left (a x^2+1\right )^{\frac {1}{2}-\frac {b}{4 a}} \left (c_1 P_{\frac {\sqrt {a^2-2 (b+2 c) a+b^2}-a}{2 a}}^{\frac {b}{2 a}-1}\left (i \sqrt {a} x\right )+c_2 Q_{\frac {\sqrt {a^2-2 (b+2 c) a+b^2}-a}{2 a}}^{\frac {b}{2 a}-1}\left (i \sqrt {a} x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.123 (sec), leaf count = 124
\[ \left \{ y \left ( x \right ) = \left ( a{x}^{2}+1 \right ) ^{{\frac {2\,a-b}{4\,a}}} \left ( {\it LegendreP} \left ( {\frac {1}{2\,a} \left ( \sqrt {{a}^{2}+ \left ( -2\,b-4\,c \right ) a+{b}^{2}}-a \right ) },{\frac {2\,a-b}{2\,a}},\sqrt {-a}x \right ) {\it \_C1}+{\it LegendreQ} \left ( {\frac {1}{2\,a} \left ( \sqrt {{a}^{2}+ \left ( -2\,b-4\,c \right ) a+{b}^{2}}-a \right ) },{\frac {2\,a-b}{2\,a}},\sqrt {-a}x \right ) {\it \_C2} \right ) \right \} \] Mathematica raw input
DSolve[c*y[x] + b*x*y'[x] + (1 + a*x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (1 + a*x^2)^(1/2 - b/(4*a))*(C[1]*LegendreP[(-a + Sqrt[a^2 + b^2 - 2*a*(b + 2*c)])/(2*a), -1 + b/(2*a), I*Sqrt[a]*x] + C[2]*LegendreQ[(-a + Sqrt[a^2 + b^2 - 2*a*(b + 2*c)])/(2*a), -1 + b/(2*a), I*Sqrt[a]*x])}}
Maple raw input
dsolve((a*x^2+1)*diff(diff(y(x),x),x)+b*x*diff(y(x),x)+c*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (a*x^2+1)^(1/4/a*(2*a-b))*(LegendreP(1/2/a*((a^2+(-2*b-4*c)*a+b^2)^(1/2)-a),1/2/a*(2*a-b),(-a)^(1/2)*x)*_C1+LegendreQ(1/2/a*((a^2+(-2*b-4*c)*a+b^2)^(1/2)-a),1/2/a*(2*a-b),(-a)^(1/2)*x)*_C2)