Find the particular solution of the differential equation ` x(dy)/(dx) - y = (x + 1 ) e^(-x) ` given that `y = 0 ` when ` x = 1 `.

Explanation

To solve the differential equation \( x \frac{dy}{dx} - y = (x + 1)e^{-x} \) with the initial condition \( y(1) = 0 \), we can follow these steps:

### Step 1: Rearranging the equation
We start by rearranging the given equation into the standard form of a linear differential equation:

\[ \frac{dy}{dx} - \frac{y}{x} = \frac{(x + 1)e^{-x}}{x} \]
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