Link to actual problem [11093] \[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+{\mathrm e}^{2 \lambda x} c -\frac {\lambda ^{2}}{4}\right ) y=0} \]
type detected by program
{"unknown"}
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 &= \operatorname {hypergeom}\left (\left [\frac {1}{4}+\frac {i \left (4 a c -b^{2}\right )}{16 \lambda \,a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 \,{\mathrm e}^{\lambda x} a +b \right )^{2}}{4 \lambda \,a^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {i {\mathrm e}^{2 \lambda x} a +\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{\lambda x}}{2 \lambda \sqrt {a}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {i {\mathrm e}^{2 \lambda x} a +\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{\lambda x}}{2 \lambda \sqrt {a}}} y}{\operatorname {hypergeom}\left (\left [\frac {4 \lambda \,a^{\frac {3}{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 \,{\mathrm e}^{\lambda x} a +b \right )^{2}}{4 \lambda \,a^{\frac {3}{2}}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [\frac {3}{4}+\frac {i \left (4 a c -b^{2}\right )}{16 \lambda \,a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 \,{\mathrm e}^{\lambda x} a +b \right )^{2}}{4 \lambda \,a^{\frac {3}{2}}}\right ) \left (2 a \,{\mathrm e}^{-\frac {i {\mathrm e}^{2 \lambda x} a -\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{\lambda x}}{2 \lambda \sqrt {a}}}+b \,{\mathrm e}^{-\frac {i {\mathrm e}^{2 \lambda x} a +\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{\lambda x}}{2 \lambda \sqrt {a}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {i {\mathrm e}^{2 \lambda x} a -\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{\lambda x}}{2 \lambda \sqrt {a}}} {\mathrm e}^{\frac {i {\mathrm e}^{2 \lambda x} a +\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{\lambda x}}{2 \lambda \sqrt {a}}} y}{\operatorname {hypergeom}\left (\left [\frac {12 \lambda \,a^{\frac {3}{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 \,{\mathrm e}^{\lambda x} a +b \right )^{2}}{4 \lambda \,a^{\frac {3}{2}}}\right ) \left (2 a \,{\mathrm e}^{\frac {i {\mathrm e}^{2 \lambda x} a +\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{\lambda x}}{2 \lambda \sqrt {a}}}+b \,{\mathrm e}^{\frac {i {\mathrm e}^{2 \lambda x} a -\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{\lambda x}}{2 \lambda \sqrt {a}}}\right )}\right ] \\ \end{align*}