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Unformatted text preview: Motivation Introduction Second Order ODEs Miscellaneous Ordinary Differential Equations Milinda Lakkam Institute for Computational and Mathematical Engineering Stanford University September 21, 2011 Motivation Introduction Second Order ODEs Miscellaneous What is an ODE? In general, an n th order ODE can be written as F ( x , y , y , y 00 , . . . , y ( n ) ) = 0 We shall assume that the differential equations can be solved explicitly for y ( n ) in terms of the remaining qunatities y ( n ) = f ( x , y , y , . . . , y ( n 1) ) A differential equation is said to be linear if it is linear in y and all its derivatives. Thus, an n th order ODE can be written as P n [ y ] = p ( x ) y ( n ) + + p n ( x ) y = r ( x ) If r ( x ) = 0, it is called homogenous . Motivation Introduction Second Order ODEs Miscellaneous What is a solution to the ODE? A solution of an n t h order ODE on some open interval I is a function y = h ( x ) that is defined and n times differentiable on I and the ODE becomes an identity when y = h ( x ). Motivation Introduction Second Order ODEs Miscellaneous Superposition principle Consider the homogeneous linear ODE, y ( n ) + p n 1 ( x ) y ( n 1) + . . . + p 1 ( x ) y (1) + p ( x ) = 0 Superposition principle: If y 1 , . . . , y m are solutions of the homogenous linear ODE, any linear combination of y 1 , . . . , y m is also a solution of the ODE. alert true only for homogeneous linear ODEs, not for nonhomogenous ODEs Motivation Introduction Second Order ODEs Miscellaneous Linear Independence and dependence Consider n functions y 1 , . . . , y n on some interval I . These functions are linearly indpendent on I if the equation k 1 y 1 ( x ) + . . . + k n y n ( x ) = 0 on I = k 1 = 0 , . . . , k n = 0 Motivation Introduction Second Order ODEs Miscellaneous Definition A general solution of the homogenous linear ODE on any open interval I is a solution of on I of the form y ( x ) = c 1 y 1 ( x ) + . . . + c n y n ( x ) where y 1 , . . . , y n is a basis of solutions on I . Initial value problem (IVP) for the n th order homogenous ODE consists of n initial conditions y ( x ) = k , y (1) ( x ) = k 1 , . . . , y ( n 1) ( x ) = k n 1 . Motivation Introduction Second Order ODEs Miscellaneous Exitence and Uniqueness Theorem for IVPs Theorem If the coefficients p ( x ) , . . . , p ( n 1) ( x ) of the homogenous linear ODE are contiuous on some open interval I and x is in I , then the IVP of the ODE has a unique solution on I. Motivation Introduction Second Order ODEs Miscellaneous Linear Dependence and Independence of solutions Consider the homogeneous linear ODE, y ( n ) + p n 1 ( x ) y ( n 1) + . . . + p 1 ( x ) y (1) + p ( x ) = 0 Let the ODE have continuous coefficients p ( x ) , . . . , p ( n 1) ( x ) on an open interval I ....
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This note was uploaded on 10/01/2011 for the course EE 221 taught by Professor Ee221a during the Spring '08 term at University of California, Berkeley.
 Spring '08
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