<< Chapter < Page Chapter >> Page >
  • Identify the order of a differential equation.
  • Explain what is meant by a solution to a differential equation.
  • Distinguish between the general solution and a particular solution of a differential equation.
  • Identify an initial-value problem.
  • Identify whether a given function is a solution to a differential equation or an initial-value problem.

Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y = f ( x ) and its derivative, known as a differential equation . Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.

Techniques for solving differential equations can take many different forms, including direct solution, use of graphs, or computer calculations. We introduce the main ideas in this chapter and describe them in a little more detail later in the course. In this section we study what differential equations are, how to verify their solutions, some methods that are used for solving them, and some examples of common and useful equations.

General differential equations

Consider the equation y = 3 x 2 , which is an example of a differential equation because it includes a derivative. There is a relationship between the variables x and y : y is an unknown function of x . Furthermore, the left-hand side of the equation is the derivative of y . Therefore we can interpret this equation as follows: Start with some function y = f ( x ) and take its derivative. The answer must be equal to 3 x 2 . What function has a derivative that is equal to 3 x 2 ? One such function is y = x 3 , so this function is considered a solution to a differential equation    .

Definition

A differential equation    is an equation involving an unknown function y = f ( x ) and one or more of its derivatives. A solution to a differential equation is a function y = f ( x ) that satisfies the differential equation when f and its derivatives are substituted into the equation.

Go to this website to explore more on this topic.

Some examples of differential equations and their solutions appear in [link] .

Examples of differential equations and their solutions
Equation Solution
y = 2 x y = x 2
y + 3 y = 6 x + 11 y = e −3 x + 2 x + 3
y 3 y + 2 y = 24 e −2 x y = 3 e x 4 e 2 x + 2 e −2 x

Note that a solution to a differential equation is not necessarily unique, primarily because the derivative of a constant is zero. For example, y = x 2 + 4 is also a solution to the first differential equation in [link] . We will return to this idea a little bit later in this section. For now, let’s focus on what it means for a function to be a solution to a differential equation.

Verifying solutions of differential equations

Verify that the function y = e −3 x + 2 x + 3 is a solution to the differential equation y + 3 y = 6 x + 11 .

To verify the solution, we first calculate y using the chain rule for derivatives. This gives y = −3 e −3 x + 2 . Next we substitute y and y into the left-hand side of the differential equation:

( −3 e −2 x + 2 ) + 3 ( e −2 x + 2 x + 3 ) .

The resulting expression can be simplified by first distributing to eliminate the parentheses, giving

−3 e −2 x + 2 + 3 e −2 x + 6 x + 9 .

Combining like terms leads to the expression 6 x + 11 , which is equal to the right-hand side of the differential equation. This result verifies that y = e −3 x + 2 x + 3 is a solution of the differential equation.

Got questions? Get instant answers now!
Got questions? Get instant answers now!
Practice Key Terms 8

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 2' conversation and receive update notifications?

Ask