##### 4.10.28 $$\left (e^{-x} x-2 y(x)\right ) y'(x)=2 e^{-2 x} x-\left (-2 y(x)+e^{-x} x+e^{-x}\right ) y(x)$$

ODE
$\left (e^{-x} x-2 y(x)\right ) y'(x)=2 e^{-2 x} x-\left (-2 y(x)+e^{-x} x+e^{-x}\right ) y(x)$ ODE Classiﬁcation

[[_Abel, 2nd type, class B]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.559476 (sec), leaf count = 81

$\left \{\left \{y(x)\to \frac {1}{2} e^{-2 x} \left (e^x x-\sqrt {e^{2 x} \left (-3 x^2+4 c_1\right )}\right )\right \},\left \{y(x)\to \frac {1}{2} e^{-2 x} \left (e^x x+\sqrt {e^{2 x} \left (-3 x^2+4 c_1\right )}\right )\right \}\right \}$

Maple
cpu = 0.053 (sec), leaf count = 88

$\left [y \left (x \right ) = -\frac {\left (-x \,{\mathrm e}^{x}+\sqrt {-3 x^{2} {\mathrm e}^{2 x}+4 \,{\mathrm e}^{2 x} \textit {\_C1}}\right ) {\mathrm e}^{-2 x}}{2}, y \left (x \right ) = \frac {{\mathrm e}^{-2 x} \left (x \,{\mathrm e}^{x}+\sqrt {-3 x^{2} {\mathrm e}^{2 x}+4 \,{\mathrm e}^{2 x} \textit {\_C1}}\right )}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (E^x*x - Sqrt[E^(2*x)*(-3*x^2 + 4*C[1])])/(2*E^(2*x))}, {y[x] -> (E^x*
x + Sqrt[E^(2*x)*(-3*x^2 + 4*C[1])])/(2*E^(2*x))}}

Maple raw input

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

Maple raw output

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