Step 1: Identify the given differential equation as linear:

$$\frac{dy}{dx} + 2y = \sin x$$

Here, $P = 2$ and $Q = \sin x$.

Step 2: Find the integrating factor (I.F.):

$$\text{I.F.} = e^{\int P \, dx} = e^{\int 2 \, dx} = e^{2x}$$

Step 3: The solution is given by:

$$y \times \text{I.F.} = \int (Q \times \text{I.F.}) \, dx + C$$

so,

$$y \times e^{2x} = \int e^{2x} \sin x \, dx + C$$

Step 4: Using integration by parts:

$$\int e^{2x} \sin x \, dx = \frac{e^{2x}(2 \sin x - \cos x)}{5}$$

Step 5: Therefore,

$$y \times e^{2x} = \frac{e^{2x}(2 \sin x - \cos x)}{5} + C$$

Step 6: Dividing both sides by $e^{2x}$, we get:

$$y = \frac{1}{5}(2 \sin x - \cos x) + Ce^{-2x}$$

∴ $y = \frac{1}{5}(2 \sin x - \cos x) + Ce^{-2x}$, which is the required general solution.