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 = 0.0353197 (sec), leaf count = 81

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

Maple
cpu = 0.049 (sec), leaf count = 26

\[ \left \{ - \left (y \relax (x ) \right ) ^{2}{{\rm e}^{2\,x}}+xy \relax (x ) {{\rm e}^{x}}-{x}^{2}+{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

-y(x)^2*exp(2*x)+x*y(x)*exp(x)-x^2+_C1 = 0