Answer
$y = \sin(2t) - 2\cos(2t) + Ce^{-t}$
Work Step by Step
We solve:
$$y' + y = 5\sin(2t)$$
Integrating factor:
$$\mu(t) = e^{\int 1dt} = e^t$$
Multiply through:
$$e^t y' + e^t y = 5 e^t \sin(2t)$$
$$\frac{d}{dt}(e^t y) = 5 e^t \sin(2t)$$
Integrate:
$$e^t y = 5 \int e^t \sin(2t)dt$$
$$\int e^t \sin(2t)dt = \frac{e^t(\sin(2t) - 2\cos(2t))}{5} + C$$
Substitute back:
$$e^t y = e^t(\sin(2t) - 2\cos(2t)) + C$$
$$y = \sin(2t) - 2\cos(2t) + C e^{-t}$$
