4.13.19 $(x^2+2xy(x)-y(x{)}^2)y^{\prime }(x)+x^2-2xy(x)+y(x{)}^2=0$

ODE
$$(x^2+2xy(x)-y(x{)}^2)y^{\prime }(x)+x^2-2xy(x)+y(x{)}^2=0$$ ODE Classification

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

Book solution method
Homogeneous equation

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

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

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

$$\{-\mathit{\_}\mathit{C}\mathit{1}+{\int }^{\frac{y\left(x\right)}{x}}\frac{{\mathit{\_}\mathit{a}}^2-2\mathit{\_}\mathit{a}-1}{{\mathit{\_}\mathit{a}}^3-3{\mathit{\_}\mathit{a}}^2+\mathit{\_}\mathit{a}-1}d\mathit{\_}\mathit{a}+\ln \left(x\right)=0\}$$ 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