##### 4.20.32 $$-x^2-2 x y(x) y'(x)+y(x)^2 y'(x)^2+2 y(x)^2=0$$

ODE
$-x^2-2 x y(x) y'(x)+y(x)^2 y'(x)^2+2 y(x)^2=0$ ODE Classiﬁcation

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

Book solution method
Change of variable

Mathematica
cpu = 3.10834 (sec), leaf count = 301

$\left \{\text {Solve}\left [\log (x)+\frac {1}{2} \left (\log \left (\frac {y(x)^2}{x^2}+1\right )-\frac {2 i \sqrt {\frac {y(x)+x}{x}} \sqrt {\frac {y(x)}{x}-1} \tan ^{-1}\left (\frac {\sqrt {\frac {y(x)^2}{x^2}-1}}{\sqrt {2}}\right )}{\sqrt {\frac {y(x)^2}{x^2}-1}}-\frac {i \sqrt {2} \sqrt {\frac {y(x)-x}{y(x)+x}} (y(x)+x)}{x}+i \sqrt {\frac {y(x)+x}{x}} \sqrt {\frac {2 y(x)}{x}-2}\right )=c_1,y(x)\right ],\text {Solve}\left [\log (x)+\frac {1}{2} \left (\log \left (\frac {y(x)^2}{x^2}+1\right )+\frac {2 i \sqrt {\frac {y(x)+x}{x}} \sqrt {\frac {y(x)}{x}-1} \tan ^{-1}\left (\frac {\sqrt {\frac {y(x)^2}{x^2}-1}}{\sqrt {2}}\right )}{\sqrt {\frac {y(x)^2}{x^2}-1}}+\frac {i \sqrt {2} \sqrt {\frac {y(x)-x}{y(x)+x}} (y(x)+x)}{x}-i \sqrt {\frac {y(x)+x}{x}} \sqrt {\frac {2 y(x)}{x}-2}\right )=c_1,y(x)\right ]\right \}$

Maple
cpu = 3.434 (sec), leaf count = 107

$\left [y \left (x \right ) = x, y \left (x \right ) = -x, y \left (x \right ) = \sqrt {-2 \sqrt {2}\, x \textit {\_C1} -\textit {\_C1}^{2}-x^{2}}, y \left (x \right ) = \sqrt {2 \sqrt {2}\, x \textit {\_C1} -\textit {\_C1}^{2}-x^{2}}, y \left (x \right ) = -\sqrt {-2 \sqrt {2}\, x \textit {\_C1} -\textit {\_C1}^{2}-x^{2}}, y \left (x \right ) = -\sqrt {2 \sqrt {2}\, x \textit {\_C1} -\textit {\_C1}^{2}-x^{2}}\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[Log[x] + (Log[1 + y[x]^2/x^2] - (I*Sqrt[2]*Sqrt[(-x + y[x])/(x + y[x])]*(
x + y[x]))/x + I*Sqrt[(x + y[x])/x]*Sqrt[-2 + (2*y[x])/x] - ((2*I)*ArcTan[Sqrt[-
1 + y[x]^2/x^2]/Sqrt[2]]*Sqrt[(x + y[x])/x]*Sqrt[-1 + y[x]/x])/Sqrt[-1 + y[x]^2/
x^2])/2 == C[1], y[x]], Solve[Log[x] + (Log[1 + y[x]^2/x^2] + (I*Sqrt[2]*Sqrt[(-
x + y[x])/(x + y[x])]*(x + y[x]))/x - I*Sqrt[(x + y[x])/x]*Sqrt[-2 + (2*y[x])/x]
 + ((2*I)*ArcTan[Sqrt[-1 + y[x]^2/x^2]/Sqrt[2]]*Sqrt[(x + y[x])/x]*Sqrt[-1 + y[x
]/x])/Sqrt[-1 + y[x]^2/x^2])/2 == C[1], y[x]]}

Maple raw input

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

Maple raw output

[y(x) = x, y(x) = -x, y(x) = (-2*2^(1/2)*x*_C1-_C1^2-x^2)^(1/2), y(x) = (2*2^(1/
2)*x*_C1-_C1^2-x^2)^(1/2), y(x) = -(-2*2^(1/2)*x*_C1-_C1^2-x^2)^(1/2), y(x) = -(
2*2^(1/2)*x*_C1-_C1^2-x^2)^(1/2)]