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The given equation is:

\[
x \, dy - (y + 1) \, dx = 0
\]

This is a first-order differential equation. To solve it, let's follow the steps:

### Step 1: Rewrite the equation
We can rewrite the equation as:

\[
x \, dy = (y + 1) \, dx
\]

Now, divide both sides by \(x(y + 1)\):

\[
\frac{dy}{y + 1} = \frac{dx}{x}
\]

### Step 2: Integrate both sides
We can now integrate both sides:

\[
\int \frac{dy}{y + 1} = \int \frac{dx}{x}
\]

The integral of \(\frac{1}{y+1}\) is \(\ln|y+1|\), and the integral of \(\frac{1}{x}\) is \(\ln|x|\), so we get:

\[
\ln|y + 1| = \ln|x| + C
\]

where \(C\) is the constant of integration.

### Step 3: Exponentiate both sides
To eliminate the logarithms, exponentiate both sides:

\[
|y + 1| = e^{\ln|x| + C} = |x| e^C
\]

Let \(e^C\) be a new constant, say \(A\). Hence, we have:

\[
|y + 1| = A|x|
\]

### Step 4: Solve for \(y\)
Finally, solving for \(y\), we get:

\[
y + 1 = \pm A x
\]

Thus,

\[
y = \pm A x - 1
\]

This is the general solution to the given differential equation.

---

Would you like further details or clarification on any step?

Here are 5 related questions to expand your understanding:
1. How do you handle first-order linear differential equations in general?
2. What is the role of the constant of integration, and why is it necessary?
3. How do logarithmic properties help in solving differential equations?
4. Can we determine the specific value of \(A\) if an initial condition is given?
5. What are the different types of first-order differential equations?

**Tip:** When integrating both sides of a differential equation, always remember to include the constant of integration.

Solve the differential equation: x dy - (y + 1) dx = 0.

Learn how to solve the first-order differential equation x dy - (y + 1) dx = 0 using separation of variables and logarithmic integration. This detailed solution covers integration of logarithmic functions and exponentiation steps, suitable for undergraduate calculus students.

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