Economics Dynamics Problems 79

Economics Dynamics Problems 79 - particular solution is y (...

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Continuous dynamic systems 63 so the particular solution is y ( t ) = 3 e 2 t + 13 te 2 t = (3 + 13 t ) e 2 t 2.8.3 Complex conjugate ( b 2 < 4 ac ) If the auxiliary equation has complex conjugate roots r and s where r = α + i β and s = α i β then e α t cos( β t ) and e α t sin( β t ) are linearly independent solutions to the second-order homogeneous equation (see exercise 10). The general solution is y ( t ) = c 1 e α t cos( β t ) + c 2 e α t sin( β t ) where c 1 and c 2 are arbitrary constants. If y (0) and y ± (0) are the initial conditions when t = 0, then we can solve for c 1 and c 2 y (0) = c 1 cos(0) + c 2 sin(0) = c 1 y ± ( t ) = ( α c 1 + β c 2 ) e α t cos( β t ) + ( α c 2 β c 1 ) e α t sin( β t ) y ± (0) = ( α c 1 + β c 2 ) e 0 cos(0) + ( α c 2 β c 1 ) e 0 sin(0) = α c 1 + β c 2 i.e. c 1 = y (0) and c 2 = y ± (0) α y (0) β Hence, the
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Unformatted text preview: particular solution is y ( t ) = y (0) e t cos( t ) + y (0) y (0) e t sin( t ) Example 2.19 y ( t ) + 2 y ( t ) + 2 y ( t ) = , y (0) = 2 and y (0) = 1 The auxiliary equation is x 2 + 2 x + 2 = with complex conjugate roots r = 2 + 4 4(2) 2 = 1 + i s = 2 4 4(2) 2 = 1 i The general solution is y ( t ) = c 1 e t cos( t ) + c 2 e t sin( t )...
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