##### 4.17.43 $$4 y'(x)^2+2 e^{2 x-2 y(x)} y'(x)-e^{2 x-2 y(x)}=0$$

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

[[_homogeneous, class C], _dAlembert]

Book solution method
Change of variable

Mathematica
cpu = 0.878773 (sec), leaf count = 153

$\left \{\text {Solve}\left [y(x)=\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \tanh ^{-1}\left (\frac {e^x}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}+c_1,y(x)\right ],\text {Solve}\left [y(x)+\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \tanh ^{-1}\left (\frac {e^x}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}=c_1,y(x)\right ]\right \}$

Maple
cpu = 10.438 (sec), leaf count = 137

$\left [y \left (x \right ) = \textit {\_C1} -\arctanh \left (\frac {1}{\RootOf \left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\RootOf \left (16 \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +\textit {\_C1} \right )\right ) {\mathrm e}^{2 \textit {\_Z}}+8 \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +\textit {\_C1} \right )\right ) {\mathrm e}^{\textit {\_Z}}+\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +\textit {\_C1} \right )-4 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right ), y \left (x \right ) = \textit {\_C1} +\arctanh \left (\frac {1}{\RootOf \left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\RootOf \left (16 \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +\textit {\_C1} \right )\right ) {\mathrm e}^{2 \textit {\_Z}}+8 \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +\textit {\_C1} \right )\right ) {\mathrm e}^{\textit {\_Z}}+\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +\textit {\_C1} \right )-4 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right )\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[y[x] == (Sqrt[E^(4*x) + 4*E^(2*(x + y[x]))]*ArcTanh[E^x/Sqrt[E^(2*x) + 4*
E^(2*y[x])]])/(E^x*Sqrt[E^(2*x) + 4*E^(2*y[x])]) + C[1], y[x]], Solve[(Sqrt[E^(4
*x) + 4*E^(2*(x + y[x]))]*ArcTanh[E^x/Sqrt[E^(2*x) + 4*E^(2*y[x])]])/(E^x*Sqrt[E
^(2*x) + 4*E^(2*y[x])]) + y[x] == C[1], y[x]]}

Maple raw input

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

Maple raw output

[y(x) = _C1-arctanh(1/RootOf(_Z^2-4*exp(RootOf(16*tanh(-1/2*_Z-x+_C1)^2*exp(_Z)^
2+8*tanh(-1/2*_Z-x+_C1)^2*exp(_Z)+tanh(-1/2*_Z-x+_C1)^2-4*exp(_Z)-1))-1)), y(x)
= _C1+arctanh(1/RootOf(_Z^2-4*exp(RootOf(16*tanh(-1/2*_Z-x+_C1)^2*exp(_Z)^2+8*ta
nh(-1/2*_Z-x+_C1)^2*exp(_Z)+tanh(-1/2*_Z-x+_C1)^2-4*exp(_Z)-1))-1))]