1. **Problem Statement:** Solve the differential equation $$\frac{dy}{dt} = -\frac{y}{t} + 2$$ for the general solution. 2. **Identify the type of equation:** This is a first-order linear differential equation of the form $$\frac{dy}{dt} + P(t)y = Q(t)$$ where $$P(t) = \frac{1}{t}$$ (note the sign change when rearranged) and $$Q(t) = 2$$. 3. **Rewrite the equation:** Move all terms to standard form: $$\frac{dy}{dt} + \frac{1}{t}y = 2$$ 4. **Find the integrating factor (IF):** The integrating factor is given by $$\mu(t) = e^{\int P(t) dt} = e^{\int \frac{1}{t} dt} = e^{\ln|t|} = |t|$$ Assuming $$t > 0$$ for simplicity, $$\mu(t) = t$$. 5. **Multiply the entire differential equation by the integrating factor:** $$t \frac{dy}{dt} + t \cdot \frac{1}{t} y = 2t$$ which simplifies to $$t \frac{dy}{dt} + y = 2t$$ 6. **Recognize the left side as a derivative:** $$\frac{d}{dt}(ty) = 2t$$ 7. **Integrate both sides with respect to $$t$$:** $$\int \frac{d}{dt}(ty) dt = \int 2t dt$$ $$ty = t^2 + C$$ 8. **Solve for $$y$$:** $$y = \frac{t^2 + C}{t} = t + \frac{C}{t}$$ **Final answer:** $$\boxed{y = t + \frac{C}{t}}$$ This solution shows how to use an integrating factor to solve a linear first-order differential equation by converting it into an exact derivative, integrating, and then solving for the dependent variable.