Link to actual problem [9817] \[ \boxed {x^{2} y^{\prime \prime \prime }+3 y^{\prime \prime } x +\left (4 a^{2} x^{2 a}+1-4 \nu ^{2} a^{2}\right ) y^{\prime }-4 a^{3} x^{2 a -1} y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_3rd_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 &= \operatorname {hypergeom}\left (\left [-\frac {1}{2}\right ], \left [\nu +1, -\nu +1\right ], -x^{2 a}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {1}{2}\right ], \left [\nu +1, -\nu +1\right ], -x^{2 a}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-2 a \nu } \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\nu \right ], \left [1-2 \nu , -\nu +1\right ], -x^{2 a}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2 a \nu } y}{\operatorname {hypergeom}\left (\left [-\frac {1}{2}-\nu \right ], \left [1-2 \nu , -\nu +1\right ], -x^{2 a}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{2 a \nu } \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\nu \right ], \left [2 \nu +1, \nu +1\right ], -x^{2 a}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-2 a \nu } y}{\operatorname {hypergeom}\left (\left [-\frac {1}{2}+\nu \right ], \left [2 \nu +1, \nu +1\right ], -x^{2 a}\right )}\right ] \\ \end{align*}