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 = 0.216729 (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 = 0.481 (sec), leaf count = 143
\[\left [y \left (x \right ) = \textit {\_C1} \left (a \,x^{2}+1\right )^{\frac {2 a -b}{4 a}} \LegendreP \left (\frac {\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-a}{2 a}, \frac {2 a -b}{2 a}, x \sqrt {-a}\right )+\textit {\_C2} \left (a \,x^{2}+1\right )^{\frac {2 a -b}{4 a}} \LegendreQ \left (\frac {\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-a}{2 a}, \frac {2 a -b}{2 a}, x \sqrt {-a}\right )\right ]\] 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))
Maple raw output
[y(x) = _C1*(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),x*(-a)^(1/2))+_C2*(a*x^2+1)^(1/4/a*(2*a-b))*LegendreQ(1/2/a*((a^2+(-2*b-4*c)*a+b^2)^(1/2)-a),1/2/a*(2*a-b),x*(-a)^(1/2))]