ODE
\[ y'(x)^2-2 y'(x)-y(x)^2=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.0633867 (sec), leaf count = 73
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {$\#$1}^2+1}}{\text {$\#$1}}-\frac {1}{\text {$\#$1}}+\sinh ^{-1}(\text {$\#$1})\& \right ]\left [c_1-x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {$\#$1}^2+1}}{\text {$\#$1}}+\frac {1}{\text {$\#$1}}+\sinh ^{-1}(\text {$\#$1})\& \right ]\left [c_1+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.029 (sec), leaf count = 85
\[ \left \{ x- \left ( y \left ( x \right ) \right ) ^{-1}-{\frac {1}{y \left ( x \right ) } \left ( 1+ \left ( y \left ( x \right ) \right ) ^{2} \right ) ^{{\frac {3}{2}}}}+y \left ( x \right ) \sqrt {1+ \left ( y \left ( x \right ) \right ) ^{2}}+{\it Arcsinh} \left ( y \left ( x \right ) \right ) -{\it \_C1}=0,x- \left ( y \left ( x \right ) \right ) ^{-1}+{\frac {1}{y \left ( x \right ) } \left ( 1+ \left ( y \left ( x \right ) \right ) ^{2} \right ) ^{{\frac {3}{2}}}}-y \left ( x \right ) \sqrt {1+ \left ( y \left ( x \right ) \right ) ^{2}}-{\it Arcsinh} \left ( y \left ( x \right ) \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[-y[x]^2 - 2*y'[x] + y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[ArcSinh[#1] - #1^(-1) - Sqrt[1 + #1^2]/#1 & ][-x + C[1]]}, {y[x] -> InverseFunction[ArcSinh[#1] + #1^(-1) - Sqrt[1 + #1^2]/#1 & ][x + C[1]]}}
Maple raw input
dsolve(diff(y(x),x)^2-2*diff(y(x),x)-y(x)^2 = 0, y(x),'implicit')
Maple raw output
x-1/y(x)-1/y(x)*(1+y(x)^2)^(3/2)+y(x)*(1+y(x)^2)^(1/2)+arcsinh(y(x))-_C1 = 0, x-1/y(x)+1/y(x)*(1+y(x)^2)^(3/2)-y(x)*(1+y(x)^2)^(1/2)-arcsinh(y(x))-_C1 = 0