ODE
\[ -\left (x^2-x y(x)-2 y(x)^2\right ) y'(x)+(y(x)+x)^2 y'(x)^2-(x-y(x)) y(x)=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
Book solution method
No Missing Variables ODE, Solve for \(y'\)
Mathematica ✓
cpu = 0.206022 (sec), leaf count = 97
\[\left \{\left \{y(x)\to -x-\sqrt {x^2+e^{2 c_1}}\right \},\left \{y(x)\to -x+\sqrt {x^2+e^{2 c_1}}\right \},\left \{y(x)\to -x-\sqrt {2 x^2+e^{2 c_1}}\right \},\left \{y(x)\to -x+\sqrt {2 x^2+e^{2 c_1}}\right \}\right \}\]
Maple ✓
cpu = 1.215 (sec), leaf count = 85
\[\left [y \left (x \right ) = -x -\sqrt {x^{2}+2 \textit {\_C1}}, y \left (x \right ) = -x +\sqrt {x^{2}+2 \textit {\_C1}}, y \left (x \right ) = \frac {-x \textit {\_C1} -\sqrt {2 \textit {\_C1}^{2} x^{2}+1}}{\textit {\_C1}}, y \left (x \right ) = \frac {-x \textit {\_C1} +\sqrt {2 \textit {\_C1}^{2} x^{2}+1}}{\textit {\_C1}}\right ]\] Mathematica raw input
DSolve[-((x - y[x])*y[x]) - (x^2 - x*y[x] - 2*y[x]^2)*y'[x] + (x + y[x])^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -x - Sqrt[E^(2*C[1]) + x^2]}, {y[x] -> -x + Sqrt[E^(2*C[1]) + x^2]}, {y[x] -> -x - Sqrt[E^(2*C[1]) + 2*x^2]}, {y[x] -> -x + Sqrt[E^(2*C[1]) + 2*x^2]}}
Maple raw input
dsolve((x+y(x))^2*diff(y(x),x)^2-(x^2-x*y(x)-2*y(x)^2)*diff(y(x),x)-(x-y(x))*y(x) = 0, y(x))
Maple raw output
[y(x) = -x-(x^2+2*_C1)^(1/2), y(x) = -x+(x^2+2*_C1)^(1/2), y(x) = (-x*_C1-(2*_C1^2*x^2+1)^(1/2))/_C1, y(x) = (-x*_C1+(2*_C1^2*x^2+1)^(1/2))/_C1]