##### 4.17.42 $$4 y'(x)^2+2 x e^{-2 y(x)} y'(x)-e^{-2 y(x)}=0$$

ODE
$4 y'(x)^2+2 x e^{-2 y(x)} y'(x)-e^{-2 y(x)}=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of variable

Mathematica
cpu = 0.624147 (sec), leaf count = 103

$\left \{\left \{y(x)\to \log \left (-e^{\frac {c_1}{2}} \sqrt {-x+e^{c_1}}\right )\right \},\left \{y(x)\to \log \left (e^{\frac {c_1}{2}} \sqrt {-x+e^{c_1}}\right )\right \},\left \{y(x)\to \log \left (-e^{\frac {c_1}{2}} \sqrt {x+e^{c_1}}\right )\right \},\left \{y(x)\to \log \left (e^{\frac {c_1}{2}} \sqrt {x+e^{c_1}}\right )\right \}\right \}$

Maple
cpu = 14.465 (sec), leaf count = 122

$\left [y \left (x \right ) = -\frac {\ln \left (-\frac {4}{x^{2}}\right )}{2}, y \left (x \right ) = \textit {\_C1} -\arctanh \left (\frac {x}{\RootOf \left (\textit {\_Z}^{2}-x^{2}-4 \,{\mathrm e}^{\RootOf \left (x^{2} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+\textit {\_C1} \right )\right )+4 \,{\mathrm e}^{\textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+\textit {\_C1} \right )\right )-x^{2}\right )}\right )}\right ), y \left (x \right ) = \textit {\_C1} +\arctanh \left (\frac {x}{\RootOf \left (\textit {\_Z}^{2}-x^{2}-4 \,{\mathrm e}^{\RootOf \left (x^{2} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+\textit {\_C1} \right )\right )+4 \,{\mathrm e}^{\textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+\textit {\_C1} \right )\right )-x^{2}\right )}\right )}\right )\right ]$ Mathematica raw input

DSolve[-E^(-2*y[x]) + (2*x*y'[x])/E^(2*y[x]) + 4*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> Log[-(E^(C[1]/2)*Sqrt[E^C[1] - x])]}, {y[x] -> Log[E^(C[1]/2)*Sqrt[E^C
[1] - x]]}, {y[x] -> Log[-(E^(C[1]/2)*Sqrt[E^C[1] + x])]}, {y[x] -> Log[E^(C[1]/
2)*Sqrt[E^C[1] + x]]}}

Maple raw input

dsolve(4*diff(y(x),x)^2+2*x*exp(-2*y(x))*diff(y(x),x)-exp(-2*y(x)) = 0, y(x))

Maple raw output

[y(x) = -1/2*ln(-4/x^2), y(x) = _C1-arctanh(1/RootOf(_Z^2-x^2-4*exp(RootOf(x^2*t
anh(-1/2*_Z+_C1)^2+4*exp(_Z)*tanh(-1/2*_Z+_C1)^2-x^2)))*x), y(x) = _C1+arctanh(1
/RootOf(_Z^2-x^2-4*exp(RootOf(x^2*tanh(-1/2*_Z+_C1)^2+4*exp(_Z)*tanh(-1/2*_Z+_C1
)^2-x^2)))*x)]