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The given equation is:

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

This is a first-order linear differential equation. To solve it, we can use the method of **integrating factors**. Here's the step-by-step approach:

### Step 1: Rewrite the equation in standard form
The standard form of a first-order linear differential equation is:

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

Rewriting the given equation:

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

Here, \( P(x) = 3 \) and \( Q(x) = e^x \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:

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

Since \( P(x) = 3 \):

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

### Step 3: Multiply both sides by the integrating factor
Multiplying the entire equation by \( e^{3x} \):

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

Simplifying the right-hand side:

\[
e^{3x} \frac{dy}{dx} + 3e^{3x}y = e^{4x}
\]

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

\[
\frac{d}{dx}(y \cdot e^{3x}) = e^{4x}
\]

### Step 4: Integrate both sides
Integrating both sides with respect to \( x \):

\[
y \cdot e^{3x} = \int e^{4x} \, dx
\]

The integral of \( e^{4x} \) is:

\[
y \cdot e^{3x} = \frac{e^{4x}}{4} + C
\]

### Step 5: Solve for \( y \)
Finally, solve for \( y \) by dividing both sides by \( e^{3x} \):

\[
y = \frac{e^{4x}}{4e^{3x}} + \frac{C}{e^{3x}} = \frac{e^x}{4} + Ce^{-3x}
\]

### Final Solution:
The general solution to the differential equation is:

\[
y = \frac{e^x}{4} + Ce^{-3x}
\]

Would you like more details or have any further questions?

### Related questions:
1. How do you solve non-linear first-order differential equations?
2. What is the method of separation of variables in solving differential equations?
3. How do you determine a particular solution given initial conditions?
4. What are homogeneous and non-homogeneous differential equations?
5. Can you explain the method of undetermined coefficients for solving non-homogeneous equations?

### Tip:
When solving first-order linear differential equations, always check if the equation is already in standard form before applying the integrating factor method.

Explore a comprehensive guide to solving the linear first-order differential equation dy/dx = e^x - 3y using the method of integrating factors. This tutorial covers necessary mathematical concepts like exponential functions and integrating factors, suitable for university-level students.

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