##### 4.19.12 $$x^2 y'(x)^2-4 x (y(x)+2) y'(x)+4 y(x) (y(x)+2)=0$$

ODE
$x^2 y'(x)^2-4 x (y(x)+2) y'(x)+4 y(x) (y(x)+2)=0$ ODE Classiﬁcation

[_separable]

Book solution method
No Missing Variables ODE, Solve for $$y'$$

Mathematica
cpu = 0.261369 (sec), leaf count = 59

$\left \{\left \{y(x)\to e^{-c_1} x \left (x-2 \sqrt {2} e^{\frac {c_1}{2}}\right )\right \},\left \{y(x)\to e^{c_1} x^2-2 \sqrt {2} e^{\frac {c_1}{2}} x\right \}\right \}$

Maple
cpu = 6.807 (sec), leaf count = 265

$\left [y \left (x \right ) = -2, y \left (x \right ) = \frac {\left (-\frac {2 \sqrt {2}\, \sqrt {x^{2} \textit {\_C1}}}{x^{2}}+1\right ) x^{2}}{\textit {\_C1}}, y \left (x \right ) = \frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{2} \textit {\_C1}}}{x^{2}}+1\right ) x^{2}}{\textit {\_C1}}, y \left (x \right ) = -\frac {\frac {16 \textit {\_C1}^{3} \left (\sqrt {2}\, x -4 \textit {\_C1} \right )}{8 \textit {\_C1}^{2}-x^{2}}+8 \textit {\_C1}^{2}-\frac {2 \textit {\_C1} \left (\sqrt {2}\, x -4 \textit {\_C1} \right ) x^{2}}{8 \textit {\_C1}^{2}-x^{2}}-x^{2}}{\textit {\_C1}^{2} \left (\frac {4 \textit {\_C1} \left (\sqrt {2}\, x -4 \textit {\_C1} \right )}{8 \textit {\_C1}^{2}-x^{2}}+1\right )}, y \left (x \right ) = -\frac {-\frac {16 \textit {\_C1}^{3} \left (\sqrt {2}\, x +4 \textit {\_C1} \right )}{8 \textit {\_C1}^{2}-x^{2}}+8 \textit {\_C1}^{2}+\frac {2 \textit {\_C1} \left (\sqrt {2}\, x +4 \textit {\_C1} \right ) x^{2}}{8 \textit {\_C1}^{2}-x^{2}}-x^{2}}{\textit {\_C1}^{2} \left (-\frac {4 \textit {\_C1} \left (\sqrt {2}\, x +4 \textit {\_C1} \right )}{8 \textit {\_C1}^{2}-x^{2}}+1\right )}\right ]$ Mathematica raw input

DSolve[4*y[x]*(2 + y[x]) - 4*x*(2 + y[x])*y'[x] + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (x*(-2*Sqrt[2]*E^(C[1]/2) + x))/E^C[1]}, {y[x] -> -2*Sqrt[2]*E^(C[1]/2
)*x + E^C[1]*x^2}}

Maple raw input

dsolve(x^2*diff(y(x),x)^2-4*x*(2+y(x))*diff(y(x),x)+4*(2+y(x))*y(x) = 0, y(x))

Maple raw output

[y(x) = -2, y(x) = (-2*2^(1/2)/x^2*(x^2*_C1)^(1/2)+1)*x^2/_C1, y(x) = (2*2^(1/2)
/x^2*(x^2*_C1)^(1/2)+1)*x^2/_C1, y(x) = -(16*_C1^3*(2^(1/2)*x-4*_C1)/(8*_C1^2-x^
2)+8*_C1^2-2*_C1*(2^(1/2)*x-4*_C1)/(8*_C1^2-x^2)*x^2-x^2)/_C1^2/(4*_C1*(2^(1/2)*
x-4*_C1)/(8*_C1^2-x^2)+1), y(x) = -(-16*_C1^3*(2^(1/2)*x+4*_C1)/(8*_C1^2-x^2)+8*
_C1^2+2*_C1*(2^(1/2)*x+4*_C1)/(8*_C1^2-x^2)*x^2-x^2)/_C1^2/(-4*_C1*(2^(1/2)*x+4*
_C1)/(8*_C1^2-x^2)+1)]