4.19.12 $x^2{y}^{\prime }(x{)}^2-4x(y(x)+2)y^{\prime }(x)+4y(x)(y(x)+2)=0$

ODE
$$x^2{y}^{\prime }(x{)}^2-4x(y(x)+2)y^{\prime }(x)+4y(x)(y(x)+2)=0$$ ODE Classification

[_separable]

Book solution method
No Missing Variables ODE, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.0719412 (sec), leaf count = 59

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

Maple ✓
cpu = 0.055 (sec), leaf count = 83

$$\{\ln \left(x\right)-\frac{1}{2}\ln (\sqrt{2y\left(x\right)+4}+2)+\frac{1}{2}\ln (\sqrt{2y\left(x\right)+4}-2)-\frac{\ln \left(y\left(x\right)\right)}{2}-\mathit{\_}\mathit{C}\mathit{1}=0,\ln \left(x\right)+\frac{1}{2}\ln (\sqrt{2y\left(x\right)+4}+2)-\frac{1}{2}\ln (\sqrt{2y\left(x\right)+4}-2)-\frac{\ln \left(y\left(x\right)\right)}{2}-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=-2\}$$ 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