Link to actual problem [7633] \[ \boxed {2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 y^{\prime } x^{3}+\left (3 x^{2}+1\right ) y=0} \]
type detected by program
{"kovacic", "second_order_bessel_ode"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {x}}{\left (x^{2}+2\right )^{\frac {3}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+2\right )^{\frac {3}{4}} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {\sqrt {x}\, \left (-\ln \left (1-\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{\frac {1}{4}}}{2}\right )+\ln \left (1+\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{\frac {1}{4}}}{2}\right )-2 \arctan \left (\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{\frac {1}{4}}}{2}\right )\right )}{\left (2 x^{2}+4\right )^{\frac {3}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x^{2}+4\right )^{\frac {3}{4}} y}{\sqrt {x}\, \left (\ln \left (1-\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{\frac {1}{4}}}{2}\right )-\ln \left (1+\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{\frac {1}{4}}}{2}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{\frac {1}{4}}}{2}\right )\right )}\right ] \\ \end{align*}