Link to actual problem [10825] \[ \boxed {y^{\prime \prime }-\left (a x +b \right ) y=0} \]
type detected by program
{"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 &= \operatorname {AiryAi}\left (-\frac {-x a -b}{\left (-a \right )^{\frac {2}{3}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {AiryAi}\left (\frac {x a +b}{\left (-a \right )^{\frac {2}{3}}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {AiryBi}\left (-\frac {-x a -b}{\left (-a \right )^{\frac {2}{3}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {AiryBi}\left (\frac {x a +b}{\left (-a \right )^{\frac {2}{3}}}\right )}\right ] \\ \end{align*}