##### 4.18.4 $$x y'(x)^2+4 y'(x)-2 y(x)=0$$

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

[_rational, _dAlembert]

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

Mathematica
cpu = 31.1537 (sec), leaf count = 49

$\text {Solve}\left [\left \{x K[1]+4=\frac {2 y(K[1])}{K[1]},y(x)=\frac {K[1] (4 K[1] \log (K[1])+c_1 K[1]+8)}{(K[1]-2)^2}\right \},\{y(x),K[1]\}\right ]$

Maple
cpu = 0.857 (sec), leaf count = 64

$\left [y \left (x \right ) = \frac {x \,{\mathrm e}^{2 \RootOf \left (-x \,{\mathrm e}^{2 \textit {\_Z}}+4 x \,{\mathrm e}^{\textit {\_Z}}-4 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_C1} +8 \textit {\_Z} -4 x \right )}}{2}+2 \,{\mathrm e}^{\RootOf \left (-x \,{\mathrm e}^{2 \textit {\_Z}}+4 x \,{\mathrm e}^{\textit {\_Z}}-4 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_C1} +8 \textit {\_Z} -4 x \right )}\right ]$ Mathematica raw input

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

Mathematica raw output

Solve[{4 + x*K[1] == (2*y[K[1]])/K[1], y[x] == (K[1]*(8 + C[1]*K[1] + 4*K[1]*Log
[K[1]]))/(-2 + K[1])^2}, {y[x], K[1]}]

Maple raw input

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

Maple raw output

[y(x) = 1/2*x*exp(2*RootOf(-x*exp(2*_Z)+4*x*exp(_Z)-4*exp(_Z)+_C1+8*_Z-4*x))+2*e
xp(RootOf(-x*exp(2*_Z)+4*x*exp(_Z)-4*exp(_Z)+_C1+8*_Z-4*x))]