Link to actual problem [6493] Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-3 x y^{\prime }+y \left (x -1\right )=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
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 &= x^{\frac {2}{3}-\frac {\sqrt {249}\, 2^{{1}/{3}} \left (4241+237 \sqrt {249}\right )^{{2}/{3}}}{4000}+\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{24000}+\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{1200}-\frac {\sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{400}} \operatorname {hypergeom}\left (\left [\right ], \left [1+\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}-\frac {3 \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}+\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}-\frac {3 \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}+\frac {79 i \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{32000000}-\frac {4241 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{96000000}, 1+\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}-\frac {3 \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}+\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}-\frac {3 \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}-\frac {79 i \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{32000000}+\frac {4241 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{96000000}\right ], -x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {\sqrt {249}\, 2^{{1}/{3}} \left (31972162+2010234 \sqrt {249}\right )^{{1}/{3}}}{4000}} x^{-\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{24000}} x^{-\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{1200}} x^{\frac {\sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{400}} y}{x^{{2}/{3}} \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {79 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \left (-\frac {4241}{237}+\sqrt {249}\right ) \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{32000000}+\frac {\left (79-3 \sqrt {249}\right ) \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}+\frac {\left (79-3 \sqrt {249}\right ) \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}, 1+\frac {79 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \left (-\frac {4241}{237}+\sqrt {249}\right ) \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{32000000}+\frac {\left (79-3 \sqrt {249}\right ) \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}+\frac {\left (79-3 \sqrt {249}\right ) \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}\right ], -x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {2}{3}-\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{48000}+\frac {\sqrt {249}\, 2^{{1}/{3}} \left (4241+237 \sqrt {249}\right )^{{2}/{3}}}{8000}-\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{2400}+\frac {\sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}-\frac {237 i \sqrt {704006+39342 \sqrt {249}-1660 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+16600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}\, 2^{{1}/{3}} \left (4241+237 \sqrt {249}\right )^{{2}/{3}}}{16000000}+\frac {4241 i \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}\, 2^{{1}/{3}} \left (4241+237 \sqrt {249}\right )^{{2}/{3}}}{48000000}} \operatorname {hypergeom}\left (\left [\right ], \left [-\frac {79 i \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{16000000}+\frac {4241 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{48000000}+1, 1-\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}+\frac {3 \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}-\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}+\frac {3 \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}-\frac {79 i \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{32000000}+\frac {4241 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{96000000}\right ], -x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{48000}} x^{-\frac {\sqrt {249}\, \left (63944324+4020468 \sqrt {249}\right )^{{1}/{3}}}{8000}} x^{\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{2400}} x^{-\frac {\sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}} x^{\frac {237 i \sqrt {704006+39342 \sqrt {249}-1660 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+16600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}\, \left (63944324+4020468 \sqrt {249}\right )^{{1}/{3}}}{16000000}} x^{-\frac {4241 i \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}\, \left (63944324+4020468 \sqrt {249}\right )^{{1}/{3}}}{48000000}} y}{x^{{2}/{3}} \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {79 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \left (-\frac {4241}{237}+\sqrt {249}\right ) \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{16000000}, 1-\frac {79 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \left (-\frac {4241}{237}+\sqrt {249}\right ) \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{32000000}+\frac {\left (3 \sqrt {249}-79\right ) \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}+\frac {\left (3 \sqrt {249}-79\right ) \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}\right ], -x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {2}{3}-\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{48000}+\frac {\sqrt {249}\, 2^{{1}/{3}} \left (4241+237 \sqrt {249}\right )^{{2}/{3}}}{8000}-\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{2400}+\frac {\sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}+\frac {237 i \sqrt {704006+39342 \sqrt {249}-1660 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+16600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}\, 2^{{1}/{3}} \left (4241+237 \sqrt {249}\right )^{{2}/{3}}}{16000000}-\frac {4241 i \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}\, 2^{{1}/{3}} \left (4241+237 \sqrt {249}\right )^{{2}/{3}}}{48000000}} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {79 i \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{16000000}-\frac {4241 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{48000000}+1, 1-\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}+\frac {3 \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}-\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}+\frac {3 \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}+\frac {79 i \sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{32000000}-\frac {4241 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{96000000}\right ], -x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {79 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{48000}} x^{-\frac {\sqrt {249}\, \left (63944324+4020468 \sqrt {249}\right )^{{1}/{3}}}{8000}} x^{\frac {79 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{2400}} x^{-\frac {\sqrt {249}\, \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}} x^{-\frac {237 i \sqrt {704006+39342 \sqrt {249}-1660 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+16600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}\, \left (63944324+4020468 \sqrt {249}\right )^{{1}/{3}}}{16000000}} x^{\frac {4241 i \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}\, \left (63944324+4020468 \sqrt {249}\right )^{{1}/{3}}}{48000000}} y}{x^{{2}/{3}} \operatorname {hypergeom}\left (\left [\right ], \left [1+\frac {79 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \left (-\frac {4241}{237}+\sqrt {249}\right ) \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{16000000}, 1+\frac {79 i \left (16964+948 \sqrt {249}\right )^{{2}/{3}} \left (-\frac {4241}{237}+\sqrt {249}\right ) \sqrt {25446+1422 \sqrt {249}-60 \left (16964+948 \sqrt {249}\right )^{{2}/{3}}+600 \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}}{32000000}+\frac {\left (3 \sqrt {249}-79\right ) \left (16964+948 \sqrt {249}\right )^{{2}/{3}}}{16000}+\frac {\left (3 \sqrt {249}-79\right ) \left (16964+948 \sqrt {249}\right )^{{1}/{3}}}{800}\right ], -x \right )}\right ] \\ \end{align*}