Find the integrating factor for the differential equation e∫P(x) dx

The integrating factor for a first-order linear differential equation is a function that, when multiplied by the entire equation, allows the equation to be rewritten in a form where the left-hand side is the derivative of a product. This enables easier integration.

Given the differential equation in standard form:

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

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

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

### Steps to Find the Integrating Factor

1. **Identify \( P(x) \):** The function \( P(x) \) is the coefficient of \( y \) in the differential equation. It’s the function that multiplies \( y \) in the equation.

2. **Integrate \( P(x) \):** Calculate the indefinite integral \( \int P(x) \, dx \).

3. **Exponentiate the Integral:** Take the exponential of the result from the integral to find the integrating factor.

### Example

Suppose the differential equation is:

\[
\frac{dy}{dx} + 2xy = e^x
\]

Here, \( P(x) = 2x \).

1. **Integrate \( P(x) \):**

\[
\int 2x \, dx = x^2 + C
\]

Since the constant of integration \( C \) will cancel out when exponentiated, it is typically omitted:

\[
\int 2x \, dx = x^2
\]

2. **Exponentiate the Integral:**

\[
\mu(x) = e^{x^2}
\]

Thus, the integrating factor is:

\[
\mu(x) = e^{x^2}
\]

### Summary

To find the integrating factor \( \mu(x) \) for a first-order linear differential equation:

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

Where \( P(x) \) is the coefficient of \( y \) in the differential equation.

Would you like more examples or further clarification on this topic?

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**Related Questions:**

1. How do you solve a linear first-order differential equation using an integrating factor?
2. What happens if the function \( P(x) \) is a constant?
3. How does the integrating factor change if the differential equation is non-linear?
4. Can integrating factors be used in higher-order differential equations?
5. What is the significance of the constant of integration when finding the integrating factor?

**Tip:** Always simplify the expression for \( P(x) \) before integrating to find the most straightforward integrating factor.

Learn how to find the integrating factor for a differential equation using the formula \( \mu(x) = e^{\int P(x) \, dx} \). Understand the concept of integrating factors and their role in solving first-order linear differential equations. Suitable for college-level students and anyone interested in differential equations and mathematical modeling.

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