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

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

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of variable

Mathematica ✓
cpu = 0.254795 (sec), leaf count = 56

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

Maple ✓
cpu = 0.733 (sec), leaf count = 83

$$\{\ln \left(x\right)-\frac{\ln \left({\mathrm{e}}^{-2y\left(x\right)}{x}^2\right)}{2}-\frac{{\mathrm{e}}^{2y\left(x\right)}}{x^2}\sqrt{(4\frac{{\mathrm{e}}^{2y\left(x\right)}}{x^2}+1)x^4{\mathrm{e}}^{-4y\left(x\right)}}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\frac{1}{\sqrt{4\frac{{\mathrm{e}}^{2y\left(x\right)}}{x^2}+1}}\right)\frac{1}{\sqrt{4\frac{{\mathrm{e}}^{2y\left(x\right)}}{x^2}+1}}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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