There are several methods for solving first-order linear equations. Here's one of them:
Bring the equation into standard form, y'+p(x) y = q(x). Then find the function r(x)=e^{∫p(x)dx}. This function is called an
integrating factor. When you multiply the equation by this factor, the left-hand side will become a total derivative, which is then easy to integrate.
Your equation is already in standard form:
y'+y=xe^{-x}, with p(x)=1, q(x)=xe^{-x}.
Then r(x)=e^{∫1dx}=e^{x} is an integrating factor, so multiply the equation by e^{x}:
e^{x}y'+e^{x}y=x
The left side is the derivative of e^{x}y:
(e^{x}y)'=x
so that when you integrate both sides of the equation,
e^{x}y=x^{2}/2+c
y=(x^{2}/2)e^{-x}+ce^{-x}.
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