4.13.19 \(\left (x^2+2 x y(x)-y(x)^2\right ) y'(x)+x^2-2 x y(x)+y(x)^2=0\)

ODE
\[ \left (x^2+2 x y(x)-y(x)^2\right ) y'(x)+x^2-2 x y(x)+y(x)^2=0 \] ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0366941 (sec), leaf count = 89

\[\text {Solve}\left [c_1=\text {RootSum}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+\text {$\#$1}-1\& ,\frac {\text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )-2 \text {$\#$1} \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )-\log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{3 \text {$\#$1}^2-6 \text {$\#$1}+1}\& \right ]+\log (x),y(x)\right ]\]

Maple
cpu = 0.016 (sec), leaf count = 40

\[ \left \{ -{\it \_C1}+\int ^{{\frac {y \relax (x ) }{x}}}\!{\frac {{{\it \_a}}^{2}-2\,{\it \_a}-1}{{{\it \_a}}^{3}-3\,{{\it \_a}}^{2}+{\it \_a}-1}}{d{\it \_a}}+\ln \relax (x ) =0 \right \} \] Mathematica raw input

DSolve[x^2 - 2*x*y[x] + y[x]^2 + (x^2 + 2*x*y[x] - y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[C[1] == Log[x] + RootSum[-1 + #1 - 3*#1^2 + #1^3 & , (-Log[-#1 + y[x]/x] -
 2*Log[-#1 + y[x]/x]*#1 + Log[-#1 + y[x]/x]*#1^2)/(1 - 6*#1 + 3*#1^2) & ], y[x]]

Maple raw input

dsolve((x^2+2*x*y(x)-y(x)^2)*diff(y(x),x)+x^2-2*x*y(x)+y(x)^2 = 0, y(x),'implicit')

Maple raw output

-_C1+Intat(1/(_a^3-3*_a^2+_a-1)*(_a^2-2*_a-1),_a = y(x)/x)+ln(x) = 0