Internal problem ID [8810]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1230.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+a x y^{\prime }+\left (a -2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 36
dsolve((x^2+1)*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(a-2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x^{2}+1\right )^{1-\frac {a}{2}}+c_{2} \hypergeom \left (\left [1, \frac {a}{2}-\frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 60
DSolve[(-2 + a)*y[x] + a*x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (x^2+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (\pi c_2 \cot \left (\frac {\pi a}{2}\right )+2 c_1\right ) P_{\frac {a-4}{2}}^{\frac {a-2}{2}}(i x) \\ \end{align*}