##### 4.20.8 $$(y(x)+x) y'(x)^2+2 x y'(x)-y(x)=0$$

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

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

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

Mathematica
cpu = 0.757209 (sec), leaf count = 188

$\left \{\left \{y(x)\to -\frac {2 \sqrt {(\cosh (c_1)+\sinh (c_1)) \left (-\sqrt {3} x+\cosh (c_1)+\sinh (c_1)\right )}+\cosh (c_1)+\sinh (c_1)}{\sqrt {3}}\right \},\left \{y(x)\to \frac {2 \sqrt {(\cosh (c_1)+\sinh (c_1)) \left (-\sqrt {3} x+\cosh (c_1)+\sinh (c_1)\right )}-\cosh (c_1)-\sinh (c_1)}{\sqrt {3}}\right \},\left \{y(x)\to \frac {-2 \sqrt {(\cosh (c_1)+\sinh (c_1)) \left (\sqrt {3} x+\cosh (c_1)+\sinh (c_1)\right )}+\cosh (c_1)+\sinh (c_1)}{\sqrt {3}}\right \},\left \{y(x)\to \frac {2 \sqrt {(\cosh (c_1)+\sinh (c_1)) \left (\sqrt {3} x+\cosh (c_1)+\sinh (c_1)\right )}+\cosh (c_1)+\sinh (c_1)}{\sqrt {3}}\right \}\right \}$

Maple
cpu = 1.826 (sec), leaf count = 119

$\left [y \left (x \right ) = \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x, y \left (x \right ) = \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x, \ln \left (x \right )-\arctanh \left (\frac {2 x +y \left (x \right )}{2 x \sqrt {\frac {x^{2}+x y \left (x \right )+y \left (x \right )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-\textit {\_C1} = 0, \ln \left (x \right )+\arctanh \left (\frac {2 x +y \left (x \right )}{2 x \sqrt {\frac {x^{2}+x y \left (x \right )+y \left (x \right )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -((Cosh[C[1]] + Sinh[C[1]] + 2*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(-(Sqrt[
3]*x) + Cosh[C[1]] + Sinh[C[1]])])/Sqrt[3])}, {y[x] -> (-Cosh[C[1]] - Sinh[C[1]]
 + 2*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(-(Sqrt[3]*x) + Cosh[C[1]] + Sinh[C[1]])])/S
qrt[3]}, {y[x] -> (Cosh[C[1]] + Sinh[C[1]] - 2*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(S
qrt[3]*x + Cosh[C[1]] + Sinh[C[1]])])/Sqrt[3]}, {y[x] -> (Cosh[C[1]] + Sinh[C[1]
] + 2*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(Sqrt[3]*x + Cosh[C[1]] + Sinh[C[1]])])/Sqr
t[3]}}

Maple raw input

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

Maple raw output

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