ODE
\[ y'(x)^2=e^{4 x-2 y(x)} \left (y'(x)-1\right ) \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
Change of variable
Mathematica ✓
cpu = 1.01965 (sec), leaf count = 165
\[\left \{\text {Solve}\left [y(x)=\frac {e^{-2 x} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}} \tanh ^{-1}\left (\frac {e^{2 x}}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}\right )}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}+2 c_1,y(x)\right ],\text {Solve}\left [y(x)+\frac {e^{-2 x} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}} \tanh ^{-1}\left (\frac {e^{2 x}}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}\right )}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}=2 c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 3.276 (sec), leaf count = 259
\[\left [x -\frac {\sqrt {-\left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right ) {\mathrm e}^{8 x -4 y \left (x \right )}}\, {\mathrm e}^{-4 x +2 y \left (x \right )} \arctanh \left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}\right )}{2 \sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}+\frac {\ln \left ({\mathrm e}^{y \left (x \right )-2 x}\right )}{2}-\frac {\ln \left (2 \,{\mathrm e}^{y \left (x \right )-2 x}+1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{y \left (x \right )-2 x}-1\right )}{4}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right )}{4}-\textit {\_C1} = 0, x +\frac {\ln \left ({\mathrm e}^{y \left (x \right )-2 x}\right )}{2}-\frac {\ln \left (2 \,{\mathrm e}^{y \left (x \right )-2 x}+1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{y \left (x \right )-2 x}-1\right )}{4}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right )}{4}+\frac {\sqrt {-\left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right ) {\mathrm e}^{8 x -4 y \left (x \right )}}\, {\mathrm e}^{-4 x +2 y \left (x \right )} \arctanh \left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}\right )}{2 \sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[y'[x]^2 == E^(4*x - 2*y[x])*(-1 + y'[x]),y[x],x]
Mathematica raw output
{Solve[y[x] == (Sqrt[E^(8*x) - 4*E^(4*x + 2*y[x])]*ArcTanh[E^(2*x)/Sqrt[E^(4*x) - 4*E^(2*y[x])]])/(E^(2*x)*Sqrt[E^(4*x) - 4*E^(2*y[x])]) + 2*C[1], y[x]], Solve[(Sqrt[E^(8*x) - 4*E^(4*x + 2*y[x])]*ArcTanh[E^(2*x)/Sqrt[E^(4*x) - 4*E^(2*y[x])]])/(E^(2*x)*Sqrt[E^(4*x) - 4*E^(2*y[x])]) + y[x] == 2*C[1], y[x]]}
Maple raw input
dsolve(diff(y(x),x)^2 = exp(4*x-2*y(x))*(diff(y(x),x)-1), y(x))
Maple raw output
[x-1/2*(-(4*exp(-4*x+2*y(x))-1)*exp(8*x-4*y(x)))^(1/2)*exp(-4*x+2*y(x))/(-4*exp(-4*x+2*y(x))+1)^(1/2)*arctanh(1/(-4*exp(-4*x+2*y(x))+1)^(1/2))+1/2*ln(exp(y(x)-2*x))-1/4*ln(2*exp(y(x)-2*x)+1)-1/4*ln(2*exp(y(x)-2*x)-1)+1/4*ln(4*exp(-4*x+2*y(x))-1)-_C1 = 0, x+1/2*ln(exp(y(x)-2*x))-1/4*ln(2*exp(y(x)-2*x)+1)-1/4*ln(2*exp(y(x)-2*x)-1)+1/4*ln(4*exp(-4*x+2*y(x))-1)+1/2*(-(4*exp(-4*x+2*y(x))-1)*exp(8*x-4*y(x)))^(1/2)*exp(-4*x+2*y(x))/(-4*exp(-4*x+2*y(x))+1)^(1/2)*arctanh(1/(-4*exp(-4*x+2*y(x))+1)^(1/2))-_C1 = 0]