Internal problem ID [6783]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 51.
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}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 61
dsolve(x^2*(2+x^2)*diff(y(x),x$2)+2*x*(x^2+5)*diff(y(x),x)+2*(3-x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (x^{2}+8\right )}{x}+\frac {c_{2} \left (\left (x^{4}+8 x^{2}\right ) \arctanh \left (\frac {\sqrt {2}}{\sqrt {x^{2}+2}}\right )+\left (-x^{2}+4\right ) \sqrt {2}\, \sqrt {x^{2}+2}\right )}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.131 (sec). Leaf size: 85
DSolve[x^2*(2+x^2)*y''[x]+2*x*(x^2+5)*y'[x]+2*(3-x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {-2 x^2 \left (32 c_1 \left (x^2+8\right )+c_2 \sqrt {x^2+2}\right )+8 c_2 \sqrt {x^2+2}+\sqrt {2} c_2 \left (x^2+8\right ) x^2 \tanh ^{-1}\left (\frac {\sqrt {x^2+2}}{\sqrt {2}}\right )}{64 x^3} \\ \end{align*}