Internal problem ID [7300]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 580.
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 (-4 x^{2}+1\right ) y^{\prime }+\left (2 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 43
dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(1-4*x^2)*diff(y(x),x)+(1+2*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} x}{\left (x^{2}+1\right )^{\frac {3}{2}}}+\frac {c_{2} x \left (-\sqrt {x^{2}+1}+\arctanh \left (\frac {1}{\sqrt {x^{2}+1}}\right )\right )}{\left (x^{2}+1\right )^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 45
DSolve[x^2*(1+x^2)*y''[x]-x*(1-4*x^2)*y'[x]+(1+2*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \left (c_2 \sqrt {x^2+1}-c_2 \tanh ^{-1}\left (\sqrt {x^2+1}\right )+c_1\right )}{\left (x^2+1\right )^{3/2}} \\ \end{align*}