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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:...
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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