# L62 - Fall 2003 Math 308/501502 6 Theory of Higher-Order...

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Fall 2003 Math 308/501–502 6 Theory of Higher-Order Linear ODEs 6.2 Homogeneous Linear Equations with Constant Coefﬁcients Fri, 26/Sep c ± 2003, Art Belmonte Summary A homogeneous linear differential equation of order n with real constant coefﬁcients has the form n X k = 0 a k y ( k ) ( x ) = 0. Its associated auxiliary or characteristic equation is n X k = 0 a k r k = 0. By the Fundamental Theorem of Algebra, there are n roots of this polynomial equation (accounting for multiplicities), which are real or occur in complex conjugate pairs. Here is how to ﬁnd solutions to the differential equation. Let r be a real root multiplicity m .Then x k e rx , k = 0 , 1 ,... m - 1, are m linearly independent solutions of the DE. Let r = α + i β be a complex root of the characteristic polynomial with multiplicity m . Then we have the following 2 m linearly independent solutions of the DE. x k e α x cos β x , k = 0 , 1 ,..., m - 1 ; x k e α x sin β x , k = 0 , 1 m - 1 . “Hand” Examples When necessary, resort to MATLAB for computations. Example A Show that y 1 ( x ) = e x , y 2 ( x ) = xe x ,and y 3 ( x ) = x 2 e x are linearly independent. Solution Assume these functions are linearly dependent. Then there exist constants c 1 , c 2 , c 3 , not all zero, such that c 1 y 1 + c 2 y 2 + c 3 y 3 = 0 for all x . In particular, for x =- 1 , 0 , 1, we have e - 1 - e - 1 e - 1 100 eee c 1 c 2 c 3 = 0 0 0 or Mc = 0 , whence c = M \ 0 = [0 ; 0 ; 0], a contradiction (since not all the c k are zero). Hence it must be the case that all the

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## This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

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L62 - Fall 2003 Math 308/501502 6 Theory of Higher-Order...

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