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 = 0.0493886 (sec), leaf count = 97
\[\left \{\left \{y(x)\to -\sqrt {e^{2 c_1}+x^2}-x\right \},\left \{y(x)\to \sqrt {e^{2 c_1}+x^2}-x\right \},\left \{y(x)\to -\sqrt {e^{2 c_1}+2 x^2}-x\right \},\left \{y(x)\to \sqrt {e^{2 c_1}+2 x^2}-x\right \}\right \}\]
Maple ✓
cpu = 0.033 (sec), leaf count = 48
\[ \left \{ x+{\frac {y \left ( x \right ) }{2}}-{\frac {{\it \_C1}}{y \left ( x \right ) }}=0,-{\frac {1}{2}\ln \left ( {\frac {-{x}^{2}+2\,xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}} \right ) }-\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] 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),'implicit')
Maple raw output
x+1/2*y(x)-1/y(x)*_C1 = 0, -1/2*ln((-x^2+2*x*y(x)+y(x)^2)/x^2)-ln(x)-_C1 = 0