Link to actual problem [12742] \[ \boxed {y^{\prime }-\frac {\sqrt {x^{2}+4 y}}{2}=-\frac {x}{2}} \] With initial conditions \begin {align*} \left [y \left (1\right ) = -{\frac {1}{4}}\right ] \end {align*}
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _Clairaut]
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 &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {x}{2}\right ] \\ \left [R &= y+\frac {x^{2}}{4}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {y}{x^{2}}, S \left (R \right ) &= 2 \ln \left (x \right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {x^{2}+4 y}}, S \left (R \right ) &= \frac {\sqrt {x^{2}+4 y}\, \sqrt {\frac {4 y^{2}}{x^{2}+4 y}+x^{2}}-2 y}{2 y x}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\left (x^{2} \left (x^{2}+4 y\right )\right )^{\frac {1}{4}}}, S \left (R \right ) &= \int _{}^{y}\frac {1}{\textit {\_a} \left (\frac {\left (-\frac {2 y^{2} \textit {\_a}}{\sqrt {x^{2} \left (x^{2}+4 y\right )}}+\sqrt {\frac {4 y^{4} \textit {\_a}^{2}}{x^{2} \left (x^{2}+4 y\right )}+\textit {\_a}^{4}}\right ) \sqrt {x^{2} \left (x^{2}+4 y\right )}}{y^{2}}+2 \textit {\_a} \right )}d \textit {\_a}\right ] \\ \end{align*}
My program’s symgen result This shows my program’s found \(\xi ,\eta \) and the corresponding ODE in canonical coordinates \(R,S\).\begin{align*} \xi &= 0 \\ \eta &=\sqrt {x^{2}+4 y} \\ \frac {dS}{dR} &= {\frac {1}{2}} \\ \end{align*}