Internal problem ID [6876]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 144.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+1\right ) y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 33
dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(1-2*x^2)*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} x}{\sqrt {x^{2}+1}}+\frac {c_{2} x \arctanh \left (\frac {1}{\sqrt {x^{2}+1}}\right )}{\sqrt {x^{2}+1}} \]
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 33
DSolve[x^2*(1+x^2)*y''[x]-x*(1-2*x^2)*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \left (c_1-c_2 \tanh ^{-1}\left (\sqrt {x^2+1}\right )\right )}{\sqrt {x^2+1}} \\ \end{align*}