What is the Integrating Factor of a Linear Differential Equation?

The integrating factor of a linear differential equation is a function that, when multiplied by the entire differential equation, makes it possible to integrate the equation directly.

Understanding Integrating Factors

The purpose of an integrating factor is to transform a non-exact differential equation into an exact one. This allows us to solve the differential equation using direct integration techniques. Integrating factors are most commonly used with first-order linear ordinary differential equations.

First-Order Linear Ordinary Differential Equations

A first-order linear ordinary differential equation can be written in the following standard form:

dy/dx + P(x)y = Q(x)

where P(x) and Q(x) are functions of x.

Calculating the Integrating Factor

The integrating factor, often denoted by μ(x) or IF, is calculated as follows:

μ(x) = e^{∫P(x) dx}

Where:

• e is the base of the natural logarithm.
• ∫P(x) dx is the integral of the function P(x) with respect to x.

How to Use the Integrating Factor

1. Identify P(x): Ensure your differential equation is in the standard form (dy/dx + P(x)y = Q(x)) and identify P(x).
2. Calculate the Integrating Factor: Use the formula μ(x) = e^{∫P(x) dx} to find the integrating factor.
3. Multiply the Entire Equation: Multiply both sides of the original differential equation by the integrating factor μ(x). This results in: μ(x)dy/dx + μ(x)P(x)y = μ(x)Q(x). The left-hand side should now be the derivative of the product of y and the integrating factor [d/dx(yμ(x))].
4. Integrate: Integrate both sides of the modified equation with respect to x. The left side will simplify to yμ(x). ∫ [d/dx(yμ(x))] dx = ∫ μ(x)Q(x) dx, therefore, yμ(x) = ∫ μ(x)Q(x) dx.
5. Solve for y: Solve the resulting equation for y to obtain the general solution to the differential equation: y = [∫ μ(x)Q(x) dx] / μ(x)

Example

Let's say we have the differential equation:

dy/dx + (2/x)y = x

1. Identify P(x): P(x) = 2/x
2. Calculate the Integrating Factor: μ(x) = e^{∫(2/x) dx} = e^2ln|x| = e^ln(x2) = x²
3. Multiply the Entire Equation: x²(dy/dx) + x²(2/x)y = x²(x) which simplifies to x²(dy/dx) + 2xy = x³
4. Integrate: ∫[x²(dy/dx) + 2xy] dx = ∫x³ dx, which means x²y = (x⁴/4) + C
5. Solve for y: y = (x²/4) + C/x²

Summary

The integrating factor is a crucial tool for solving first-order linear ordinary differential equations. It transforms a non-exact equation into an exact one, allowing direct integration and ultimately leading to the solution.