To solve the differential equation $\frac{dy}{dx} = 1 + \frac{2x - y}{x^2}$ with the initial condition $y(1) = 2$, let's go through the steps.

Step 1: Rewrite the Equation

We start by isolating terms involving $y$: $\frac{dy}{dx} = 1 + \frac{2x}{x^2} - \frac{y}{x^2}$ Simplify $\frac{2x}{x^2}$ to get $\frac{2}{x}$: $\frac{dy}{dx} = 1 + \frac{2}{x} - \frac{y}{x^2}$

Step 2: Rearrange to a Linear Form

Rewrite the equation as: $\frac{dy}{dx} + \frac{y}{x^2} = 1 + \frac{2}{x}$ This is now a first-order linear differential equation in the form: $\frac{dy}{dx} + P(x) y = Q(x)$ where $P(x) = \frac{1}{x^2}$ and $Q(x) = 1 + \frac{2}{x}$.

Step 3: Find the Integrating Factor

The integrating factor $I(x)$ is given by: $I(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x^2} \, dx}$ Calculating the integral: $\int \frac{1}{x^2} \, dx = -\frac{1}{x}$ Thus, the integrating factor is: $I(x) = e^{-\frac{1}{x}} = \frac{1}{e^{1/x}}$

Step 4: Multiply Through by the Integrating Factor

Multiplying both sides of the differential equation by $\frac{1}{e^{1/x}}$: $\frac{1}{e^{1/x}} \frac{dy}{dx} + \frac{y}{x^2 e^{1/x}} = \left(1 + \frac{2}{x}\right) \frac{1}{e^{1/x}}$

Step 5: Solve