Show that
a) Y= x exp-x statisfies the equation xy'=(1-x)y
Use product rule to get: $\displaystyle y = xe^{-x} \ \ \Rightarrow \ \ y' = e^{-x} - xe^{-x}$
So: $\displaystyle \text{LHS} = xy' = x \left(e^{-x} - xe^{-x}\right) = \cdots$
And: $\displaystyle \text{RHS} = (1-x)y = (1-x)(xe^{-x}) = \cdots$
Are they equal? If so, you're done.