##### 4.18.39 $$4 x y'(x)^2+2 x y'(x)-y(x)=0$$

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

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

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

Mathematica
cpu = 0.220234 (sec), leaf count = 58

$\left \{\left \{y(x)\to \frac {1}{4} e^{2 c_1} \left (-2 \sqrt {x}+e^{2 c_1}\right )\right \},\left \{y(x)\to \frac {1}{4} e^{-4 c_1} \left (1-2 e^{2 c_1} \sqrt {x}\right )\right \}\right \}$

Maple
cpu = 0.076 (sec), leaf count = 51

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

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

Mathematica raw output

{{y[x] -> (E^(2*C[1])*(E^(2*C[1]) - 2*Sqrt[x]))/4}, {y[x] -> (1 - 2*E^(2*C[1])*S
qrt[x])/(4*E^(4*C[1]))}}

Maple raw input

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

Maple raw output

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