Unformatted text preview: suppose the solution takes the form y n = c 1 r n + c 2 s n for some constants r and s and where c 1 and c 2 depend on the initial conditions y and y 1 . If this indeed is correct, then c 1 r n + 2 + c 2 s n + 2 = a ( c 1 r n + 1 + c 2 s n + 1 ) + b ( c 1 r n + c 2 s n ) Re-arranging and factorising, we obtain c 1 r n ( r 2 − ar − b ) + c 2 s n ( s 2 − as − b ) = So long as r and s are chosen to be the solution values to the general quadratic equation x 2 − ax − b = i.e. x = r and x = s , where r ±= s , then y n = c 1 r n + c 2 s n is a solution to the dynamic system. This quadratic equation is referred to as the characteristic equation of the dynamical system. If r > s , then we call y 1 = c 1 r n the dominant solution and r the dominant characteristic root. Furthermore, given we have obtained the solution values r and s , and given the initial conditions, y and y 1 , then we can solve for the two unknown coef±cients,...
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- Fall '11