The given differential equation is:

\[
\frac{dy}{dx} = -1 - 3y
\]

This is a first-order linear differential equation. To solve it, we will use the method of separation of variables or an integrating factor. Let's proceed with the integrating factor method.

### Step 1: Rewrite the equation

\[
\frac{dy}{dx} + 3y = -1
\]

This is now in the standard linear form:

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

where \( P(x) = 3 \) and \( Q(x) = -1 \).

### Step 2: Find the integrating factor

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

\[
\mu(x) = e^{\int P(x) dx} = e^{\int 3 dx} = e^{3x}
\]

### Step 3: Multiply the equation by the integrating factor

Multiplying both sides of the equation by \( e^{3x} \), we get:

\[
e^{3x} \frac{dy}{dx} + 3e^{3x} y = -e^{3x}
\]

The left-hand side is now the derivative of \( e^{3x} y \):

\[
\frac{d}{dx} \left( e^{3x} y \right) = -e^{3x}
\]

### Step 4: Integrate both sides

Integrating both sides with respect to \( x \):

\[
e^{3x} y = \int -e^{3x} dx
\]

The integral on the right side is:

\[
\int -e^{3x} dx = \frac{-e^{3x}}{3}
\]

Thus, we have:

\[
e^{3x} y = \frac{-e^{3x}}{3} + C
\]

### Step 5: Solve for \( y \)

To solve for \( y \), divide both sides by \( e^{3x} \):

\[
y = \frac{-e^{3x}}{3e^{3x}} + \frac{C}{e^{3x}} = -\frac{1}{3} + Ce^{-3x}
\]

Thus, the general solution to the differential equation is:

\[
y = -\frac{1}{3} + Ce^{-3x}
\]

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

---

Would you like more details on any step or have any questions?

Here are some related questions for further exploration:

1. What is the general form of a linear differential equation?
2. How does the integrating factor method work for solving differential equations?
3. Can this differential equation be solved using separation of variables?
4. What are the conditions for a differential equation to be classified as first-order linear?
5. How can initial conditions be applied to determine the constant \( C \)?

**Tip:** When dealing with first-order linear differential equations, the integrating factor method is a reliable approach to finding the solution!

This page provides a detailed solution for the first-order linear differential equation dy/dx = -1 - 3y. Learn how to apply the integrating factor method to find the general solution, with step-by-step guidance. Suitable for students studying undergraduate-level calculus.

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