1 / 10

A Brief Introduction to Differential Equations

A Brief Introduction to Differential Equations. Michael A. Karls. What is a differential equation?. A differential equation is an equation which involves an unknown function and some of its derivatives. Example 1: (Some differential equations). More Terminology.

cala
Download Presentation

A Brief Introduction to Differential Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. A Brief Introduction to Differential Equations Michael A. Karls

  2. What is a differential equation? • A differential equation is an equation which involves an unknown function and some of its derivatives. • Example 1: (Some differential equations)

  3. More Terminology • In an equation which involves the derivative of one variable with respect to another variable, the former is called a dependent variable and the latter an independent variable. • Any variable which is neither independent nor dependent is a parameter. • Example 2: Apply this definition to Example 1. • For (1), t is independent, P is dependent, and k is a parameter. • For (2), t is independent, x is dependent, and m, b, k, and  are parameters. • For (3), x and t are independent, u is dependent, and there are no parameters.

  4. How to solve certain differential equations • We now look at how to solve differential equations of the form:

  5. Case 1: (x,y) = f(x) • In this case we solve by integrating! • We call (6) the general solution to (5). • To find a particular solution, we need to specify some initial data such as y(x0)=y0.

  6. Case 2: (x,y) = f(x)g(y) • In this case, we say the differential equation (4) is separable. • To solve, separate variables and integrate! • Again, (7) yields a general solution to (6). • To find a particular solution, initial data needs to be specified.

  7. Remark on Case 2: • If g(y0)=0, (7) has a solution of the form y ´ y0, which will be lost in this solution process!

  8. Example 3 • Solve the initial value problem: • Solution: Use separation of variables!

  9. Solution to Example 3

  10. Solution to Example 3 (cont.) • Note that P ´ 0 is also a solution to (9). Hence the general solution is: • P(t) = Cekt, with C 2 R. For a particular solution, use (10). • P0 = P(0) = Ce0 = C, which implies P(t) = P0ekt.

More Related