Nonexact Equations: Integrating Factors

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  • 0:04 The Integrating Factor Method
  • 0:24 Nonexact Linear Equation
  • 1:23 Procedure for Solving…
  • 4:16 Example
  • 5:25 Lesson Summary
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Lesson Transcript
Instructor: Russell Frith
The method of integrating factors is a technique for solving linear, first order partial differential equations that are not exact. In this lesson, a definition is given for this type of equation and a procedure is presented for finding the solution for this type of equation.

The Integrating Factor Method

The integrating factor method is a technique used to solve linear, first-order partial differential equations of the form:


Where a(x) and b(x) are continuous functions. The method applies to such equations that are nonexact.

Nonexact Linear Equation

The general form of a partial differential equation is:


This equation may be reformulated as:


If this equation is exact, then the following equality holds:


The presence of this equality in the linear partial differential equation makes that equation exact.

The condition of exactness ensures the existence of a function F(x,y) such that:


This lesson treats finding solutions of nonexact differential equations, which means those equations will not have the properties of exact differential equations.

Procedures for Solving Equations

Let's take a moment to go over the procedure for solving these nonexact linear first order partial differential equations.

Step 1

Write the differential equation in standard form:


Step 2

Compute the integrating factor. The formula is:


The integrating factor is a function that is used to transform the differential equation into an equation that can be solved by applying the Fundamental Theorem of Calculus. Recall that the Fundamental Theorem of Calculus states that if f is a continuous function on a closed interval [a,b] then F is the antiderivative of f on [a,b].

Step 3

Multiply both sides of the equation from Step 1 by the integrating factor from Step 2:


Step 4

The left hand side of the equation in Step 3 is the derivative of:


This is after the chain rule for differentiation is applied. Recall that the chain rule for differentiation is the procedure for differentiating the composition of two continuous functions.


The equation from Step 3 may now be written as:


Step 5

Integrate both sides of the new differential equation from Step 4 with respect to x:


One may apply the Fundamental Theorem of Calculus to the left hand side of this equation to simplify it as:


Where C1 is the constant of integration.

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