Internal problem ID [8936]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1357.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}+\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 103
dsolve(diff(diff(y(x),x),x) = -1/x*(a*x^2+a-1)/(x^2+1)*diff(y(x),x)-(b*x^2+c)/x^2/(x^2+1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{1-\frac {a}{2}} \LegendreP \left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right )+c_{2} x^{1-\frac {a}{2}} \LegendreQ \left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right ) \]
✓ Solution by Mathematica
Time used: 0.353 (sec). Leaf size: 240
DSolve[y''[x] == -(((c + b*x^2)*y[x])/(x^2*(1 + x^2))) - ((-1 + a + a*x^2)*y'[x])/(x*(1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^{\frac {1}{2} \left (-\sqrt {(a-2)^2-4 c}-a+2\right )} \left (c_1 \, _2F_1\left (\frac {1}{4} \left (-\sqrt {(a-1)^2-4 b}-\sqrt {(a-2)^2-4 c}+1\right ),\frac {1}{4} \left (\sqrt {(a-1)^2-4 b}-\sqrt {(a-2)^2-4 c}+1\right );1-\frac {1}{2} \sqrt {(a-2)^2-4 c};-x^2\right )+c_2 x^{\sqrt {(a-2)^2-4 c}} \, _2F_1\left (\frac {1}{4} \left (-\sqrt {(a-1)^2-4 b}+\sqrt {(a-2)^2-4 c}+1\right ),\frac {1}{4} \left (\sqrt {(a-1)^2-4 b}+\sqrt {(a-2)^2-4 c}+1\right );\frac {1}{2} \left (\sqrt {(a-2)^2-4 c}+2\right );-x^2\right )\right ) \\ \end{align*}