Economics Dynamics Problems 130

Economics Dynamics Problems 130 - 2 2 1 2 √ 48 2 = √ 4...

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114 Economic Dynamics Complex conjugate roots 6 If the roots are complex conjugate then r = α + β i and s = α β i and R n cos( β t ) and R n sin( β t ) are solutions and the general solution is y n = c 1 R n cos( θ n ) + c 2 R n sin( θ n ) where R = ± α 2 + β 2 , cos θ = α R and sin θ = β R or tan θ = sin θ cos θ = β α Example 3.12 (complex conjugate) Consider y n + 2 4 y n + 1 + 16 y n = 0 The characteristic equation is x 2 4 x + 16 = 0 with roots r , s = 4 ± 16 64 2 = 2 ± ² 48 2 ³ i i.e. r = 2 + 1 2 48 i α = 2 s = 2 1 2 48 i β = 1 2 48 and polar coordinates R = ´
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Unformatted text preview: 2 2 + ( 1 2 √ 48) 2 = √ 4 + 12 = 4 cos θ = α R = 2 4 = 1 2 and sin θ = β R = 1 2 √ 48 4 = √ 3 2 Implying θ = π/ 3. Hence y n = c 1 4 n cos µ n π 3 ¶ + c 2 4 n sin µ n π 3 ¶ 6 In this section the complex roots are expressed in polar coordinate form (see Allen 1965 or Chiang 1984)....
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