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: $y(x) = \frac{e^x}{3} + Ce^{-2x}$

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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 $\frac{d}{dx} (\mu(x)y)$).