Link to actual problem [13694] \[ \boxed {x^{2} y^{\prime \prime }-7 y^{\prime } x +15 y=4 x^{2}+2 x +3} \]
type detected by program
{"kovacic", "second_order_euler_ode", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2", "second_order_change_of_variable_on_y_method_2"}
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*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {3 x}{8}, \underline {\hspace {1.25 ex}}\eta &= x^{2}+\frac {3}{32} x\right ] \\ \left [R &= y-\frac {4 x^{2}}{3}-\frac {x}{4}, S \left (R \right ) &= \frac {8 \ln \left (x \right )}{3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{2}, \underline {\hspace {1.25 ex}}\eta &= -\frac {1}{5}-\frac {x}{8}+y\right ] \\ \left [R &= -\frac {-20 y+4+5 x}{20 x^{2}}, S \left (R \right ) &= 2 \ln \left (x \right )\right ] \\ \end{align*}