Link to actual problem [1127] \[ \boxed {4 x y^{\prime \prime }+2 y^{\prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (\sqrt {x}\right ) \end {align*}
type detected by program
{"reduction_of_order", "second_order_bessel_ode", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2"}
type detected by Maple
[[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
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 &= {\mathrm e}^{i x \sqrt {\frac {1}{x}}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-i x \sqrt {\frac {1}{x}}} y\right ] \\ \end{align*}