This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 334 Lecture #9 3.1,2.9: Introduction to Second Order ODEs The Simplest Type of a Second Order ODE. This is the linear homogeneous with constant coefficients ODE ay 00 + by + cy = 0 . [We assume that a 6 = 0. This ODE is also autonomous the independent variable does not appear explicitly in the ODE.] To find solutions of this ODE, we apply the principle of educated guessing. Recall that solutions of the first order linear homogeneous ODE by + cy = 0 are expo nential functions. For the second order ODE we try the guess y = e rt for a constant r . To see if this guess works, we substitute it into the second order ODE: ar 2 e rt + bre rt + ce rt = 0 ( ar 2 + br + c ) e rt = 0 ar 2 + br + c = 0 . This quadratic equation in r is called the characteristic equation of the ODE [its roots give values of r for which the functions e rt are solutions.] The Case of Real and Distinct Roots. Suppose that the roots r 1 , r 2 of the characteristic equation are real and distinct. [The other two possibilities real equal roots and complex conjugate roots will be discussed in later lectures.] Two solutions of the ODE are y 1 = e r 1 t and y 2 = e r 2 t . [Now it is time to explain what linear means for an ODE.] For arbitrary constants c 1 and c 2 , the linear combination y = c 1 y 1 + c 2 y 2 = c 1 e r 1 t + c 2 e r 2 t is ALSO a solution of the ODE:...
View
Full
Document
This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
 Spring '11
 Smith
 Math

Click to edit the document details