##### 4.20.4 $$y(x) y'(x)^2-(y(x)+x) y'(x)+y(x)=0$$

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

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

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

Mathematica
cpu = 1.5692 (sec), leaf count = 181

$\left \{\text {Solve}\left [\frac {x \left (-1-i \sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {3 y(x)}{x}+1}\right )}{y(x)}+2 \left (\log \left (\frac {y(x)}{x}\right )-2 i \tan ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}-1}}{\sqrt {\frac {3 y(x)}{x}+1}}\right )+\log (x)-2 c_1\right )=0,y(x)\right ],\text {Solve}\left [\frac {x \left (-1+i \sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {3 y(x)}{x}+1}\right )}{y(x)}+2 \left (\log \left (\frac {y(x)}{x}\right )+2 i \tan ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}-1}}{\sqrt {\frac {3 y(x)}{x}+1}}\right )+\log (x)-2 c_1\right )=0,y(x)\right ]\right \}$

Maple
cpu = 0.192 (sec), leaf count = 267

$\left [y \left (x \right ) = x, \ln \left (x \right )-\frac {x \left (\frac {x^{2}+2 x y \left (x \right )-3 y \left (x \right )^{2}}{x^{2}}\right )^{\frac {3}{2}}}{2 y \left (x \right )}-\arctanh \left (\frac {x +y \left (x \right )}{x \sqrt {\frac {x^{2}+2 x y \left (x \right )-3 y \left (x \right )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )+\sqrt {\frac {x^{2}+2 x y \left (x \right )-3 y \left (x \right )^{2}}{x^{2}}}-\frac {3 y \left (x \right ) \sqrt {\frac {x^{2}+2 x y \left (x \right )-3 y \left (x \right )^{2}}{x^{2}}}}{2 x}-\frac {x}{2 y \left (x \right )}-\textit {\_C1} = 0, \ln \left (x \right )+\frac {x \left (\frac {x^{2}+2 x y \left (x \right )-3 y \left (x \right )^{2}}{x^{2}}\right )^{\frac {3}{2}}}{2 y \left (x \right )}+\arctanh \left (\frac {x +y \left (x \right )}{x \sqrt {\frac {x^{2}+2 x y \left (x \right )-3 y \left (x \right )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-\sqrt {\frac {x^{2}+2 x y \left (x \right )-3 y \left (x \right )^{2}}{x^{2}}}+\frac {3 y \left (x \right ) \sqrt {\frac {x^{2}+2 x y \left (x \right )-3 y \left (x \right )^{2}}{x^{2}}}}{2 x}-\frac {x}{2 y \left (x \right )}-\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[2*((-2*I)*ArcTan[Sqrt[-1 + y[x]/x]/Sqrt[1 + (3*y[x])/x]] - 2*C[1] + Log[x
] + Log[y[x]/x]) + (x*(-1 - I*Sqrt[-1 + y[x]/x]*Sqrt[1 + (3*y[x])/x]))/y[x] == 0
, y[x]], Solve[2*((2*I)*ArcTan[Sqrt[-1 + y[x]/x]/Sqrt[1 + (3*y[x])/x]] - 2*C[1]
+ Log[x] + Log[y[x]/x]) + (x*(-1 + I*Sqrt[-1 + y[x]/x]*Sqrt[1 + (3*y[x])/x]))/y[
x] == 0, y[x]]}

Maple raw input

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

Maple raw output

[y(x) = x, ln(x)-1/2/y(x)*x*((x^2+2*x*y(x)-3*y(x)^2)/x^2)^(3/2)-arctanh((x+y(x))
/x/((x^2+2*x*y(x)-3*y(x)^2)/x^2)^(1/2))+ln(y(x)/x)+((x^2+2*x*y(x)-3*y(x)^2)/x^2)
^(1/2)-3/2/x*y(x)*((x^2+2*x*y(x)-3*y(x)^2)/x^2)^(1/2)-1/2*x/y(x)-_C1 = 0, ln(x)+
1/2/y(x)*x*((x^2+2*x*y(x)-3*y(x)^2)/x^2)^(3/2)+arctanh((x+y(x))/x/((x^2+2*x*y(x)
-3*y(x)^2)/x^2)^(1/2))+ln(y(x)/x)-((x^2+2*x*y(x)-3*y(x)^2)/x^2)^(1/2)+3/2/x*y(x)
*((x^2+2*x*y(x)-3*y(x)^2)/x^2)^(1/2)-1/2*x/y(x)-_C1 = 0]