First-Order Linear Differential Equations

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  • 0:04 Definition and Explanation
  • 0:55 An Opening Example
  • 4:10 An Example
  • 6:11 Lesson Summary
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Lesson Transcript
Instructor: Christopher Haines
In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.

Definition and Explanation

A first-order linear differential equation is an equation of the form


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This is where


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are all functions of the independent variable x. We can alternatively write (1) as


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where


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It's important to note what the equation here is showing you: that D has the property of linearity.


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Therefore, D is a linear operator.

We're solving the equation for y with A and f as fixed functions. Examples of a first-order linear differential equation include:


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and


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An Opening Example

Consider the differential equation:


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First, observe that (2) takes the form of (1). We would like to write the left side of (2) in the form:


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This is where g is to be determined. This identity is known as the product rule for differentiation. This way, when we integrate both sides, the left side will be the integral of a derivative, which in turn is the original function yg. We do not know the function y, because it's what we are solving for. If this can be accomplished, the only integration we will need to really think about is that on the right side of (2).

However, the left side of (2) doesn't have the desired form of a derivative of a product. Thus, we introduce an integrating factor, which is a function that, upon multiplication of both sides of (2), forces the left side to be the derivative of a product of two functions. That is, we want the integrating factor h to have the following property:


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for some function g.

In the next section, we're going to learn that a formula for the integrating factor can be derived. For now, let's use the method of trial and error and suppose the integrating factor is h(x) = exp(x). Multiplying both sides of (2) by this factor produces the equation:


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Now we can rewrite the left side of (3) by applying the product rule for differentiation, so it becomes:


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Next, recognize that if we integrate the left side of (4), we obtain the original function. The right side of (4) becomes the integral of the respective function on the right side. A constant of integration is also introduced in the process. A constant of integration indicates that a function has infinitely many anti-derivatives, but all of them with the exception of a constant.

For example, an anti-derivative of the function f = 1 is F = x, while another anti-derivative is F = x + 2. C, in the equation here, is assumed here, since we haven't imposed any initial conditions.

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Before continuing, let's pause to reference an integration method we're going to frequently need in the examples to come: integration by parts. For any two functions u and v:


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Now, using integration by parts, we can evaluate the integral on the right side of the equation, as you can see here:


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Then, by substituting equation (6) back into equation (5), and then multiplying both sides by exp(-x), we obtain the general solution to (2):


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This is where K is a real number.

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