ODE
\[ \left (x^2+2 x y(x)^2-y(x)^2\right ) y'(x)-y(x)^2 \left (x^2-y(x)^2\right )+y'(x)^3-\left (y(x)^2+2 x\right ) y'(x)^2=0 \] ODE Classification
[_quadrature]
Book solution method
No Missing Variables ODE, Solve for \(y'\)
Mathematica ✓
cpu = 0.201291 (sec), leaf count = 43
\[\left \{\left \{y(x)\to -\frac {1}{x+c_1}\right \},\left \{y(x)\to x+c_1 e^{-x}-1\right \},\left \{y(x)\to -x+c_1 e^x-1\right \}\right \}\]
Maple ✓
cpu = 0.049 (sec), leaf count = 35
\[\left [y \left (x \right ) = \frac {1}{-x +\textit {\_C1}}, y \left (x \right ) = -x -1+\textit {\_C1} \,{\mathrm e}^{x}, y \left (x \right ) = x -1+{\mathrm e}^{-x} \textit {\_C1}\right ]\] Mathematica raw input
DSolve[-(y[x]^2*(x^2 - y[x]^2)) + (x^2 - y[x]^2 + 2*x*y[x]^2)*y'[x] - (2*x + y[x]^2)*y'[x]^2 + y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(x + C[1])^(-1)}, {y[x] -> -1 + x + C[1]/E^x}, {y[x] -> -1 - x + E^x*C[1]}}
Maple raw input
dsolve(diff(y(x),x)^3-(2*x+y(x)^2)*diff(y(x),x)^2+(x^2-y(x)^2+2*x*y(x)^2)*diff(y(x),x)-(x^2-y(x)^2)*y(x)^2 = 0, y(x))
Maple raw output
[y(x) = 1/(-x+_C1), y(x) = -x-1+_C1*exp(x), y(x) = x-1+exp(-x)*_C1]