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 = 0.0719412 (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 = 0.055 (sec), leaf count = 83

\[ \left \{ \ln \relax (x ) -{\frac {1}{2}\ln \left (\sqrt {2\,y \relax (x ) +4}+2 \right ) }+{\frac {1}{2}\ln \left (\sqrt {2\,y \relax (x ) +4}-2 \right ) }-{\frac {\ln \left (y \relax (x ) \right ) }{2}}-{\it \_C1}=0,\ln \relax (x ) +{\frac {1}{2}\ln \left (\sqrt {2\,y \relax (x ) +4}+2 \right ) }-{\frac {1}{2}\ln \left (\sqrt {2\,y \relax (x ) +4}-2 \right ) }-{\frac {\ln \left (y \relax (x ) \right ) }{2}}-{\it \_C1}=0,y \relax (x ) =-2 \right \} \] 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),'implicit')

Maple raw output

y(x) = -2, ln(x)-1/2*ln((2*y(x)+4)^(1/2)+2)+1/2*ln((2*y(x)+4)^(1/2)-2)-1/2*ln(y(
x))-_C1 = 0, ln(x)+1/2*ln((2*y(x)+4)^(1/2)+2)-1/2*ln((2*y(x)+4)^(1/2)-2)-1/2*ln(
y(x))-_C1 = 0