Internal problem ID [8947]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1368.
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 {x y^{\prime } a}{x^{2}+1}+\frac {b y}{\left (x^{2}+1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 81
dsolve(diff(diff(y(x),x),x) = -a*x/(x^2+1)*diff(y(x),x)-b/(x^2+1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x^{2}+1\right )^{\frac {1}{2}-\frac {a}{4}} \LegendreP \left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right )+c_{2} \left (x^{2}+1\right )^{\frac {1}{2}-\frac {a}{4}} \LegendreQ \left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 88
DSolve[y''[x] == -((b*y[x])/(1 + x^2)^2) - (a*x*y'[x])/(1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (x^2+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (c_1 P_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {(a-2)^2+4 b}}(i x)+c_2 Q_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {(a-2)^2+4 b}}(i x)\right ) \\ \end{align*}