Chapter 9: First-Order Differential Equations - Practice Exercises - Page 557: 14

$y=2xe^{x}-e^{x}+ce^{-x}$

Here, we have $y'+y=4xe^x$ The integrating factor is: $I=e^{\int (1) dx}=e^x$ Now, $e^x[y'+y]=4(xe^x) (e^x)$ This implies that $\int [e^xy]' =\int 4xe^{2x} dx$ Thus, $ye^x=2xe^{2x}-e^{2x}+c$ Hence, $y=(2x) (e^{x})-e^{x}+ce^{-x}$