${\displaystyle f'(x)+xf(x)\;=\;(x-1)e^{-x}\,}$

by superposition:

• blue: solutions of the homogeneous equation ${\displaystyle h(x)=ke^{-x^{2}}\,}$   for   ${\displaystyle k=-0.3,-0.2,-0.1,0,0.1,0.2,0.3\,}$
• green: particular solution of the inhomogeneous equation: ${\displaystyle {\bar {f}}(x)=e^{-x}\,}$
• red: solutions of the inhomogeneous equation: ${\displaystyle f(x)=h(x)+{\bar {f}}(x)=ke^{-x^{2}}+e^{-x}\,}$   for   ${\displaystyle k=-0.3,-0.2,-0.1,0,0.1,0.2,0.3\,}$