The differential equation you've provided is:

$\frac{dy}{dx} - \frac{xy}{x+1} = x$

This is a first-order linear ordinary differential equation, which can be solved using the method of integrating factors. Let's solve it step-by-step.

Step 1: Rewrite the equation in standard linear form

The standard form of a linear first-order differential equation is:

$\frac{dy}{dx} + P(x) y = Q(x)$

We can rewrite the given equation as:

$\frac{dy}{dx} = \frac{xy}{x+1} + x$

Rearrange it to:

$\frac{dy}{dx} + \left( -\frac{x}{x+1} \right) y = x$

Now we have $P(x) = -\frac{x}{x+1}$ and $Q(x) = x$.

Step 2: Find the integrating factor

The integrating factor $\mu(x)$ is given by:

$\mu(x) = e^{\int P(x) \, dx} = e^{\int -\frac{x}{x+1} \, dx}$

We need to compute the integral:

$\int -\frac{x}{x+1} \, dx$

To simplify, we use substitution: let $u = x+1$, so $du = dx$ and $x = u-1$. The integral becomes:

$\int -\frac{u-1}{u} \, du = -\int \left( 1 - \frac{1}{u} \right) \, du = -\left( u - \ln|u| \right)$

Substituting back $u = x+1$:

$\int -\frac{x}{x+1} \, dx = -(x+1 - \ln|x+1|)$

Thus, the integrating factor is:

$\mu(x) = e^{-(x+1 - \ln|x+1|)} = \frac{e^{-(x+1)}}{|x+1|}$

Step 3: Multiply the entire differential equation by the integrating factor

Multiplying both sides of the equation by $\mu(x)$, we get:

$\frac{e^{-(x+1)}}{|x+1|} \frac{dy}{dx} - \frac{e^{-(x+1)}}{|x+1|} \frac{xy}{x+1} = \frac{e^{-(x+1)}}{|x+1|} x$

This simplifies to:

$\frac{d}{dx} \left( y \cdot \frac{e^{-(x+1)}}{|x+1|} \right) = \frac{e^{-(x+1)}}{|x+1|} x$

Step 4: Solve the equation

Now integrate both sides with respect to $x$ to find $y(x)$.

Let me know if you'd like me to continue and compute the solution step-by-step or if you want further clarification.