378_pdfsam_math 54 differential equation solutions odd

378_pdfsam_math 54 - ing to(6.19 Thus s is the smallest number m such that x m cos βx and x m sin βx are not solutions to the corresponding

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Chapter 6 where Y ( x ) is the part of y h ( x ) corresponding to the roots of a n r n 2 s + ··· + a 0 2 s =0 . Since the operator ( D 2 + β 2 ) annihilates f ( x )= a cos βx + b sin βx , applying this operator to both sides in (6.19), we obtain ( D 2 + β 2 ) ( a n D n + a n 1 D n 1 + ··· + a 0 ) [ y ]=( D 2 + β 2 )[ f ( x )] = 0 . The corresponding auxiliary equation, ( r 2 + β 2 ) ( a n r n + a n 1 r n 1 + ··· + a 0 ) =0 ( r 2 + β 2 ) s +1 ( a n r n 2 s + ··· + a 0 2 s ) =0 has r = βi as its root of multiplicity s + 1. Therefore, a general solution to this equation is given by y ( x )=( c 1 cos βx + c 2 sin βx )+ x ( c 3 cos βx + c 4 sin βx ) + ··· + x s 1 ( c 2 s 1 cos βx + c 2 s sin βx )+ x s ( c 2 s +1 cos βx + c 2 s +2 sin βx )+ Y ( x ) . Since, y ( x )= y h ( x )+ y p ( x ), comparing y ( x )w ith y h ( x ) given in (6.21), we conclude that y p ( x )= x s ( c 2 s +1 cos βx + c 2 s +2 sin βx ) . All that remains is to note that, for any m<s , the functions x m cos βx and x m sin βx
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Unformatted text preview: ing to (6.19). Thus s is the smallest number m such that x m cos βx and x m sin βx are not solutions to the corresponding homogeneous equation. 39. Writing the system in operator form yields ( D 2 − 1) [ x ] + y = 0 , x + ( D 2 − 1) [ y ] = e 3 t . Subtracting the Frst equation from the second equation multiplied by ( D 2 − 1), we get n ( D 2 − 1 ) [ x ] + ( D 2 − 1 ) 2 [ y ] o − ±( D 2 − 1 ) [ x ] + y ² = ( D 2 − 1 )³ e 3 t ´ − 0 = 8 e 3 t ⇒ n ( D 2 − 1 ) 2 − 1 o [ y ] = 8 e 3 t ⇒ D 2 ( D 2 − 2 ) [ y ] = 8 e 3 t . (6.22) 374...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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