Chapter 2 - First Order Differential Equations - 2.1 Linear Equations; Method of Integrating Factors - Problems - Page 39: 13

$y=\frac{2(e^tt-e^t)+3}{e^{-t}}$

Since in form of $\frac{dy}{dt} + ay = g(t)$ $\mu(t) = e^{-t}$ Multiply by $\mu(t) $ $e^{-t}y'-e^{-t}y=2te^{2t}e^{-t}$ $\int{\frac{d}{dt}(e^{-t}y)} = \int{2e^tt}$ $e^{-t}y = 2(e^tt-e^t)+c$ $y=\frac{2(e^tt-e^t)+c}{e^{-t}}$ Plug in $t=0, y=1$ $1=\frac{2(-1)+c}{1}$ $c=3$