An **integrating factor** is a function used to solve first-order linear differential equations. It simplifies the process of solving such equations by converting them into exact equations, making them easier to integrate.

### General Form of First-Order Linear Differential Equation:
A first-order linear differential equation is typically written as:
\[
\frac{dy}{dx} + P(x)y = Q(x)
\]
Where \( P(x) \) and \( Q(x) \) are functions of \( x \).

### Steps to Solve Using the Integrating Factor:

1. **Identify the integrating factor**: 
   The integrating factor \( \mu(x) \) is given by the formula:
   \[
   \mu(x) = e^{\int P(x) dx}
   \]

2. **Multiply the entire equation by the integrating factor**:
   Multiply both sides of the original equation by \( \mu(x) \). This gives:
   \[
   \mu(x) \frac{dy}{dx} + \mu(x) P(x)y = \mu(x) Q(x)
   \]

3. **Recognize the left side as a derivative**:
   The left side of the equation should now look like the derivative of a product:
   \[
   \frac{d}{dx} \left( \mu(x) y \right) = \mu(x) Q(x)
   \]

4. **Integrate both sides**:
   Now, integrate both sides with respect to \( x \):
   \[
   \mu(x) y = \int \mu(x) Q(x) dx + C
   \]
   Where \( C \) is the constant of integration.

5. **Solve for \( y \)**:
   Finally, solve for \( y \) by dividing both sides by \( \mu(x) \):
   \[
   y(x) = \frac{1}{\mu(x)} \left( \int \mu(x) Q(x) dx + C \right)
   \]

### Example:
Solve the differential equation:
\[
\frac{dy}{dx} + 2y = e^x
\]

1. **Identify \( P(x) = 2 \)**. The integrating factor is:
   \[
   \mu(x) = e^{\int 2 dx} = e^{2x}
   \]

2. **Multiply the entire equation by \( e^{2x} \)**:
   \[
   e^{2x} \frac{dy}{dx} + 2e^{2x} y = e^{3x}
   \]

3. **Recognize the derivative**:
   \[
   \frac{d}{dx} \left( e^{2x} y \right) = e^{3x}
   \]

4. **Integrate both sides**:
   \[
   e^{2x} y = \int e^{3x} dx = \frac{e^{3x}}{3} + C
   \]

5. **Solve for \( y \)**:
   \[
   y(x) = \frac{e^{3x}/3 + C}{e^{2x}} = \frac{e^{x}}{3} + Ce^{-2x}
   \]

Thus, the solution is:
\[
y(x) = \frac{e^x}{3} + Ce^{-2x}
\]

---

Would you like more details or have any questions about the process?

### Related Questions:
1. What is an exact differential equation?
2. How do you solve a non-homogeneous differential equation using an integrating factor?
3. Can an integrating factor be used for non-linear differential equations?
4. What are the conditions under which the method of integrating factors fails?
5. How do you derive the formula for the integrating factor?

### Tip:
To quickly check if you've applied the integrating factor correctly, ensure that after multiplying, the left-hand side becomes a perfect derivative (i.e., it looks like \( \frac{d}{dx} (\mu(x)y) \)).

Learn to solve first-order linear differential equations using the integrating factor method. This guide explains key concepts like identifying the integrating factor, multiplying through, and solving for y with step-by-step examples. Ideal for students in Grades 11-12.

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