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

by superposition:

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