Internal problem ID [8805]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1225.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 24
dsolve((x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = x c_{1}+c_{2} \left (\arcsinh \relax (x ) x -\sqrt {x^{2}+1}\right ) \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 39
DSolve[y[x] - x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -c_2 \sqrt {x^2+1}+c_2 x \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )+c_1 x \\ \end{align*}