Internal problem ID [8804]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1224.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 23
dsolve((x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \left (\sqrt {a}\, \arcsinh \relax (x )\right )+c_{2} \cos \left (\sqrt {a}\, \arcsinh \relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 50
DSolve[a*y[x] + x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos \left (\sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )\right )+c_2 \sin \left (\sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )\right ) \\ \end{align*}