Unformatted text preview: ( a n D n + a n âˆ’ 1 D n âˆ’ 1 + Â·Â·Â· + a ) [ y ] = f ( x ) . (6.19) The characteristic equation, corresponding to the associated homogeneous equation, is a n r n + a n âˆ’ 1 r n âˆ’ 1 + Â·Â·Â· + a = 0 . (6.20) Suppose that r = Î²i is a root of (6.20) of multiplicity s â‰¥ 0. ( s = 0 means that r = Î²i is not a root.) Then (6.20) can be factored as a n r n + a n âˆ’ 1 r n âˆ’ 1 + Â·Â·Â· + a = ( r 2 + Î² 2 ) s ( a n r n âˆ’ 2 s + Â·Â·Â· + a /Î² 2 s ) = 0 and so a general solution to the homogeneous equation is given by y h ( x ) = ( c 1 cos Î²x + c 2 sin Î²x ) + x ( c 3 cos Î²x + c 4 sin Î²x ) + Â·Â·Â· + x s âˆ’ 1 ( c 2 s âˆ’ 1 cos Î²x + c 2 s sin Î²x ) + Y ( x ) , (6.21) 373...
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Complex number, general solution, homogeneous equation

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