4.2 - 4.2 HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS...

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4.2: HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS KIAM HEONG KWA Consider the n th order linear homogeneous equation (1) a 0 d n y dt n + a 1 d n - 1 y dt n - 1 + ··· + a n - 1 dy dt + y = 0 , where a i R , i = 0 , 2 , ··· ,n , and a 0 6 = 0. Associated with (1) is its characteristic polynomial (2) P n ( r ) = a 0 r n + a 1 r n - 1 + ··· + a n - 1 r + a n and its characteristic equation (3) P n ( r ) = 0 . The zeros of P n are called the characteristic roots of (1). We shall describe the relation between the solutions of (1) and its characteristic roots. The reader is referred to the appendix for some useful facts about the zeros of polynomials. If z R , then e zt is a solution of (1) if and only if P n ( z ) = 0 . This is so since (4) d j dt j e zt = z j e zt for each j , j = 1 , 2 , ··· ,n , and, thus, substituting e zt for y in (1) yields (5) P n ( z ) e zt = ( a 0 z n + a 1 z n - 1 + ··· + a n - 1 z + a n ) e zt = 0 . Clearly, this equation is consistent if and only if P n ( z ) = 0. If z = α + C , where α,β R with β 6 = 0, then it can also be shown that e αt cos βt and e αt sin βt are solutions of (1) if and only if P n ( z ) = 0 . On the other hand, it is a consequence of the fundamental theorem of algebra that (6) P n ( r ) = a 0 ( r - z 1 )( r - z 2 ) ··· ( r - z n ) , Date : January 26, 2011. 1
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2 KIAM HEONG KWA where z j , j = 1 , 2 , ··· ,n , are the n zeros of P n ( r ). Some of these zeros, say z j * , may have multiplicity > 1 in the sense that the factor r - z j * may appear more than once in (6). The multiplicity of z j * refers to the number of times the factor r - z j * appears on the right-hand side of (6). Hence if w k , k = 1 , 2 , ··· ,m , are the distinct zeros of P n ( r ), then (7) P n ( r ) = a 0 ( r - w 1 ) n 1 ( r - w 2 ) n 2 ··· ( r - w m ) n m , where n k is the multiplicity of w k for each k , k = 1 , 2 , ··· ,m . Let w k be a zero of P n ( r ) with multiplicity n k . If w k R , then (8) e w k t , te w k t , ··· , t n k - 1 e w k t are solutions of (1) . Clearly, for any c l R , l = 1 , 2 , ··· ,n k , (9) n k X l =1 c l t l - 1 e w k t is a solution of (1) . This is a consequence of the principle of super- position that says that any linear combination of solutions of a linear homogeneous equation is again a solution of the same equation. While if w k = α k + k C , where α k k R and β k 6 = 0 , then (10) e α k t cos( β k t ) , te α k t cos( β k t ) , ··· , t n k - 1 e α k t cos( β k t ) , e α k t sin( β k t ) , te α k t sin( β k t ) , ··· , t n k - 1 e α k t sin( β k t ) , are solutions of (1) , and so is (11) n k X l =1 t l - 1 e α k t [ A l cos( β k t ) + B l sin( β k t )] for any real constants A l and B l , l = 1 , 2 , ··· ,n l . In fact, it can be shown that the complex conjugate of w k , i.e., ¯ w k = α - , is also a zero of P n with the same multiplicity in this case. Furthermore, it is read- ily seen that ¯ w k gives rise to the same set of solutions to (1) as
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This note was uploaded on 11/11/2011 for the course MATH 255.01 taught by Professor Kwa during the Winter '11 term at Ohio State.

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4.2 - 4.2 HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS...

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