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 = 0.254795 (sec), leaf count = 56
\[\left \{\left \{y(x)\to \frac {1}{2} \log \left (\frac {1}{4} e^{c_1} \left (e^{c_1}-2 x\right )\right )\right \},\text {Solve}\left [c_1+\log \left (\sqrt {x^2+4 e^{2 y(x)}}+x\right )=2 y(x),y(x)\right ]\right \}\]
Maple ✓
cpu = 0.733 (sec), leaf count = 83
\[ \left \{ \ln \left ( x \right ) -{\frac {\ln \left ( {{\rm e}^{-2\,y \left ( x \right ) }}{x}^{2} \right ) }{2}}-{\frac {{{\rm e}^{2\,y \left ( x \right ) }}}{{x}^{2}}\sqrt { \left ( 4\,{\frac {{{\rm e}^{2\,y \left ( x \right ) }}}{{x}^{2}}}+1 \right ) {x}^{4}{{\rm e}^{-4\,y \left ( x \right ) }}}{\it Artanh} \left ( {\frac {1}{\sqrt {4\,{\frac {{{\rm e}^{2\,y \left ( x \right ) }}}{{x}^{2}}}+1}}} \right ) {\frac {1}{\sqrt {4\,{\frac {{{\rm e}^{2\,y \left ( x \right ) }}}{{x}^{2}}}+1}}}}-{\it \_C1}=0 \right \} \] 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]*(E^C[1] - 2*x))/4]/2}, Solve[C[1] + Log[x + Sqrt[4*E^(2*y[x]) + x^2]] == 2*y[x], y[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),'implicit')
Maple raw output
ln(x)-1/2*ln(exp(-2*y(x))*x^2)-((4/x^2*exp(2*y(x))+1)*x^4*exp(-4*y(x)))^(1/2)/x^2*exp(2*y(x))/(4/x^2*exp(2*y(x))+1)^(1/2)*arctanh(1/(4/x^2*exp(2*y(x))+1)^(1/2))-_C1 = 0