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 = 0.0122171 (sec), leaf count = 43
\[\left \{\left \{y(x)\to -\frac {1}{c_1+x}\right \},\left \{y(x)\to c_1 e^{-x}+x-1\right \},\left \{y(x)\to c_1 e^x-x-1\right \}\right \}\]
Maple ✓
cpu = 0.014 (sec), leaf count = 36
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{-1}+x-{\it \_C1}=0,y \left ( x \right ) =-x-1+{{\rm e}^{x}}{\it \_C1},y \left ( x \right ) =x-1+{{\rm e}^{-x}}{\it \_C1} \right \} \] 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),'implicit')
Maple raw output
1/y(x)+x-_C1 = 0, y(x) = -x-1+exp(x)*_C1, y(x) = x-1+exp(-x)*_C1